Proof. Internal symmetry gives \(\mathbb E\phi^i(f)=0\), so one-field vectors lie in \(\Omega^\perp\). Their positive energy-momentum spectral measures therefore put no weight at energies in \([0,m)\). If \(\mathop{\mathrm{supp}}\rho\) met \((0,m^2)\), some compact interval \(J\subset(0,m^2)\) would have positive \(\rho\)-measure. At sufficiently small spatial momenta, all shells with mass squared in \(J\) have energy below \(m\), a contradiction. Massless spectral weight would likewise give weight at arbitrarily small positive energies. Thus \(\mathcal M\subset[m,\infty)\). The nonzero one-field norm in Proposition 2(i) implies \(\rho\ne0\), so \(m_*<\infty\).

The Källén–Lehmann representation (Källén 1952; Lehmann 1954) and separated-time Euclidean continuation give (7). To check convergence without an assumption on the nature of the mass support, recall that a positive tempered Fourier measure has polynomial growth. Restricting its spatial momentum to \(|p|\le1\) shows that \(\rho([m^2,R^2])\) also has at most polynomial growth: the mass-shell factor \(1/(2E(p,\mu))\) on this region is bounded below by a constant times \((1+R)^{-1}\). Exponential damping therefore makes (7) and its derivatives absolutely convergent locally for \(t>0\).

Euclidean invariance rotates a separation of length \(r>0\) to \((r,0)\). At that point every integrand is positive and \(\rho\ne0\). For \(r\ge1\), factor \(e^{-m_*r}\) out of the integral and use \[e^{-r(E-m_*)}\le e^{-(E-m_*)}.\] The remaining integral is finite by the preceding growth bound. This proves (8), as well as continuity away from coincidence. All uses of the kernel in this paper are off the coincidence diagonal. ◻

Proof. Cover the support of each test by a bounded number of unit boxes. For an even moment, nonnegativity of the component moment measure allows us to replace all test factors by their absolute values and apply (6). This gives \(\mathbb E|X_f^i|^{2r}\le A^{2r}(2r)!\). Cauchy–Schwarz gives the odd absolute moments, after increasing \(A\). Summing the exponential series at any \(\kappa<A^{-1}\) proves (9). Applying the same argument to \(f-g\), with its supremum factored out, proves (10). The box bound is independent of the box positions, so none of these constants depends on translations. Products of any fixed number of such variables are controlled by Hölder’s inequality, even when the tests are translated independently. ◻

Proof. Transfer in the two coordinate directions gives \[Z_N(t,w)=\mathop{\mathrm{Tr}}K_{N,w}^{t/a}=Z_N(w,t).\] For fixed \(w\), the quantity \(P_N(t,w)\) is \((aw)^{-1}\) times the logarithm of the \(\ell^{t/a}\) norm of the eigenvalue sequence of \(K_{N,w}\). These norms decrease as their exponent increases. Symmetry gives the same monotonicity in \(w\). The finite-volume interaction is bounded in absolute value by \(2\beta_Ntw/a^2\), so the infimum is finite for each fixed \(N\). Separate monotonicity shows that \(P_N(t,w)\) tends to this infimum when both lengths tend to infinity.

On the dyadic squares, (5) gives the exact identity \[P_N(w_k,w_k)-P_N(w_{k+1},w_{k+1}) =\frac{\Delta_N(w_k)}{4w_k^2}.\] Telescoping proves \(0\le P_N(w_k,w_k)-P_{\infty,N}\le Ce^{-cw_k}\) uniformly in \(N\). For general \(t,w\) choose \(w_k\le\min(t,w)<2w_k\). Monotonicity bounds the pressure difference by its value on that square, proving (14).

At fixed \(w\), letting \(t\to\infty\) gives \(P_N(t,w)\to(aw)^{-1}\log\lambda_{0,N}(w)\), since the normalized excited trace tends to zero. Equation (14) then proves the bound for \(g_N\). Finally \[ \log R_{N,w}(t) =tw\bigl(P_N(t,w)-P_{\infty,N}\bigr)-t g_N(w). \tag{16}\] The left side is nonnegative and \(g_N(w)\ge0\), so the remaining bound follows. ◻

Proof. For sufficiently large \(w\), Equation (15) at \(t=w\) gives \[e^{-we_N(w)}\le s_{N,w}(w) \le e^{Cw^2e^{-cw}}-1\le C'w^2e^{-cw}.\] Taking logarithms yields \(e_N(w)\ge c-w^{-1}\log(C'w^2)\ge c/2\) after increasing \(w_0\). ◻

Proof. The absolute value of the insertion kernel is bounded pointwise by \(\|F\|_\infty\) times the kernel of \(T_{N,w}^{r/a}\). Apply this inequality to \(|v|\) and use \(\|T_{N,w}^{r/a}\|=1\) to obtain the operator norm bound. Put \(P_0=|\Omega_{N,w}\rangle\langle\Omega_{N,w}|\). The torus expectation is \[\frac{\mathop{\mathrm{Tr}}(A_FT_{N,w}^{(t-r)/a})}{1+s_{N,w}(t)}.\] Subtracting the ground-state term leaves a numerator bounded by \(\|F\|_\infty s_{N,w}(t-r)\), because \(T_{N,w}^{(t-r)/a}-P_0\) is positive and has that trace. The denominator contributes at most \(\|F\|_\infty s_{N,w}(t)\). This proves the estimate. ◻

Proof. First compare the tori \((w,w)\) and \((2w,w)\) with their common infinite-time cylinder. A slab enclosing the support has thickness at most \(3w/4\), up to one lattice step, so all remaining transfer lengths are at least \(w/4-a\). Equation (15) and Lemma 7 give an error \(C\|F\|_\infty e^{-cw}\); polynomial factors in \(w\) are absorbed by decreasing \(c\). Next compare \((2w,w)\) and \((2w,2w)\) in the spatial direction. The transverse circumference is now \(2w\), and the remaining transfer lengths are again at least \(w/4-a\). The same bounds apply without an aspect-ratio restriction.

Repeat these two comparisons at widths \(2w,4w,\ldots\) and sum \(C\|F\|_\infty\sum_{j\ge0}e^{-c2^jw}\). At fixed cutoff the periodic square limit is the plane state in Proposition 2. More precisely, on a fixed finite support the uniform estimate for all bounded measurable \(F\) makes the sequence of marginal laws Cauchy in total variation; its limit agrees with that periodic state on continuous functions and hence on all bounded measurable functions. This proves (18). The cylinder assertion follows directly from the first application of Lemma 7. ◻

Proof. Let \(\chi_w(x)=\max(-w,\min(x,w))\) and replace each variable \(X_j\) by \(\chi_w(X_j)\). If there are \(r\) variables, the bounded product has supremum at most \(w^r\), so Proposition 8 bounds its comparison error by \(Cw^re^{-cw}\). On either probability space, Lemma 4, Hölder’s inequality, and the exponential tail give \[\mathbb E\left|\prod_{j=1}^r X_j- \prod_{j=1}^r\chi_w(X_j)\right| \le \sum_{j=1}^r \mathbb E\!\left[\prod_{i=1}^r|X_i|\,\mathbf 1_{\{|X_j|>w\}}\right] \le Ce^{-c'w}.\] For example, Cauchy–Schwarz separates each indicator from the product, whose second moment is uniformly bounded by Hölder. Combining the estimates and reducing the exponential rate proves the claim. Only bounded insertions have been used on cylinders. ◻

Proof. Relative to the product angle reference law, the density of \(\alpha\) is proportional to \(Z_{\mathrm I}(\alpha)Z_{\mathrm R}(\alpha)\). For either conditional model, write \(O_e\) for its bond observable and \(J_e(\alpha)\) for its coupling. At distinct vertices \(x,y\), \[ \partial_x\partial_y\log Z =\sum_e\mathbb E(O_e)\,\partial_x\partial_yJ_e +\sum_{e,f}\mathop{\mathrm{Cov}}(O_e,O_f)\, \partial_xJ_e\,\partial_yJ_f\ge0. \tag{24}\] Indeed all first derivatives of the sine couplings are nonnegative and all first derivatives of the cosine couplings are nonpositive. A coupling’s only nonzero mixed derivative at distinct sites is at its endpoints, where it is nonnegative in both cases. All bond means and covariances in (24) are nonnegative by the inequalities just proved or recalled. Thus the density satisfies the FKG lattice condition, which implies association (Fortuin et al. 1971). One may restrict the angles to compact subintervals of \((0,\pi/2)\), discretize, and pass to the limit; this also handles the endpoints of the angle interval.

To obtain the same fact for the plane rotator, write its two coordinates as \((\cos\gamma\,\varepsilon,\sin\gamma\,\varepsilon')\), where \(0\le\gamma\le\pi/2\) and the two signs are conditionally independent ferromagnetic Ising systems. The calculation (24) proves association of \(\gamma\). The conditional mean of a single-component monomial is nonnegative. Including its amplitude factors, it decreases with \(\gamma\) for the first coordinate and increases for the second. Conditional independence and association therefore give nonpositive covariance between unlike components. For like components, the conditional covariance is nonnegative by GKS, and the covariance of conditional means is nonnegative by association. This proves both plane-rotator assertions.

In the three-component decomposition, the conditional mean of a color-\(3\) monomial is likewise nonnegative and increasing in \(\alpha\). The conditional mean of a color-\(1\) monomial is nonnegative and decreasing: a product of cosines is a nonnegative linear combination of \(\cos(k\cdot\vartheta)\), and both the amplitude factors and the rotator couplings decrease with \(\alpha\). Conditional independence now gives the nonpositive covariance between colors \(3\) and \(1\). For like colors, use GKS or (23) for the conditional covariance and association for the covariance of conditional means. Component symmetry gives every remaining choice of colors. ◻

Proof of Proposition 10. Expand each bond exponential in (19) into powers of the dot product. The uniform sphere moment tensor of order \(2h\) is \[\int_{S^2}q^{i_1}\cdots q^{i_{2h}}\,d\sigma(q) =\frac1{3\cdot5\cdots(2h+1)} \sum_{\pi}\prod_{\{b,c\}\in\pi}\mathbf 1_{i_b=i_c},\] where the sum is over pairings of the \(2h\) labeled factors; odd tensors vanish. Following the contracted bond lines through these site pairings pairs the external factors, with closed lines contributing positive color sums. Consequently, for an arbitrary color assignment to the combined lists \(\mathcal A,\mathcal B\), the joint moment has the form \[ \mathbb E\prod_{b=1}^{r+s}q_{x_b}^{i_b} =\sum_{\pi}d_\pi \prod_{\{b,c\}\in\pi}\mathbf 1_{i_b=i_c}, \qquad d_\pi\ge0. \tag{25}\] The coefficients do not depend on the external colors. The finite graph and finite couplings ensure absolute convergence of the exponential expansion; all compatible contracted terms are nonnegative. This justifies collecting them by their external pairing, including when sites are repeated.

Crossing pairs. Call a pair crossing if it joins the two lists. Since both list sizes are even, a pairing with a crossing pair has at least two. For any such pairing we claim \[ d_\pi\le G^2. \tag{26}\] Assign color \(3\) to the endpoints of one crossing pair, color \(2\) to the endpoints of another, and color \(1\) to all remaining endpoints. The resulting moment contains \(d_\pi\) as a nonnegative summand. Condition on \(\alpha\). In the rotator model, Lemma 11 bounds the moment of the color-\(2\) pair times the remaining color-\(1\) monomial by the product of their means, and hence by the color-\(2\) pair mean, because the color-\(1\) monomial’s mean lies in \([0,1]\). Include the factors \(\cos\alpha_x\) in this statement. The resulting pair mean is nonnegative and decreasing in \(\alpha\). Indeed, rotator invariance gives \[\mathbb E(\sin\vartheta_x\sin\vartheta_y\mid\alpha) =\tfrac12\mathbb E\bigl(\cos(\vartheta_x-\vartheta_y)\mid\alpha\bigr),\] also when \(x=y\), and both this cosine mean and the amplitude factors decrease with \(\alpha\). The color-\(3\) pair mean, including its sine factors, is nonnegative and increasing. Association bounds the average of their product above by the product of their averages. Each average is a two-point function between the lists, at most \(G\) by component symmetry. This proves (26).

Consider next single-component monomials on the two lists. Compare the case where their colors agree with the case where they differ. Their products of means agree by symmetry, while their joint moments differ by exactly the sum of the crossing contributions in (25). The two covariances have opposite weak signs by Lemma 11. Thus each absolute covariance is at most the number of pairings times \(G^2\). The same conclusion holds for any even sublists of \(\mathcal A\) and \(\mathcal B\), with the same \(G\). If either \(P\) or \(Q\) has odd degree in at least one color, its own mean vanishes. Every compatible pairing for \(PQ\) then crosses the two lists, so (26) also proves the required bound in this case.

It remains to treat monomials even in every color. This is the only case where subtracting the product of mixed-component means cannot be read directly from the pairing expansion. Write \(P=W_PV_P\), where \(W_P\) uses color \(3\) and \(V_P\) uses colors \(1,2\). Let \(V_P^*\) replace every color in \(V_P\) by color \(1\), and define \[a_P=\mathbb E(W_P\mid\alpha),\qquad b_P=\mathbb E(V_P\mid\alpha),\qquad B_P=\mathbb E(V_P^*\mid\alpha).\] Use the corresponding notation for \(Q\). We have \(0\le a_P,B_P\le1\) and \(|b_P|\le1\); \(a_P\) is increasing and \(B_P\) decreasing. Conditional independence of signs and rotator angles gives \(\mathbb E(P\mid\alpha)=a_Pb_P\). Thus the total covariance identity reads \[ \mathop{\mathrm{Cov}}(P,Q)=\mathbb E\mathop{\mathrm{Cov}}(P,Q\mid\alpha) +\mathop{\mathrm{Cov}}(a_Pb_P,a_Qb_Q). \tag{27}\] We bound the conditional fluctuations first, then the covariance of the conditional means. A Fourier comparison with the single-color monomials \(V_P^*,V_Q^*\) will control both terms.

The common Fourier comparison. Factor out the common amplitude \(A_P(\alpha)\), a product of cosines, and expand the coordinate factors: \[ V_P=A_P(\alpha)\sum_k c_k\cos(k\cdot\vartheta), \qquad V_P^*=A_P(\alpha)\sum_k d_k\cos(k\cdot\vartheta), \qquad |c_k|\le d_k. \tag{28}\] Each factor has two exponential terms with coefficients of absolute value \(1/2\). Replacing sines by cosines replaces their phases by positive coefficients. Since there are evenly many sine factors, the coefficients are real and invariant under \(k\mapsto-k\). Grouping terms at a common frequency preserves the displayed domination, even at repeated sites.

Conditional fluctuations. Conditional independence gives \[\begin{align*} \mathop{\mathrm{Cov}}(P,Q\mid\alpha) ={}&\mathop{\mathrm{Cov}}(W_P,W_Q\mid\alpha)\, \mathbb E(V_PV_Q\mid\alpha)\\ &+a_Pa_Q\mathop{\mathrm{Cov}}(V_P,V_Q\mid\alpha). \end{align*}\] The first conditional covariance is nonnegative. The Fourier comparison (28) and the entrywise nonnegative cosine covariances in (23) imply \[|\mathop{\mathrm{Cov}}(V_P,V_Q\mid\alpha)| \le\mathop{\mathrm{Cov}}(V_P^*,V_Q^*\mid\alpha).\] All remaining factors have absolute value at most one. Averaging thus bounds \(|\mathbb E\mathop{\mathrm{Cov}}(P,Q\mid\alpha)|\) by \[\mathbb E\mathop{\mathrm{Cov}}(W_P,W_Q\mid\alpha) +\mathbb E\mathop{\mathrm{Cov}}(V_P^*,V_Q^*\mid\alpha).\] Each summand is at most its full single-component covariance: the omitted covariance of conditional means is nonnegative, since both \(a\)’s increase and both \(B\)’s decrease. The single-component bounds therefore control the first term of (27) by \(C_{r,s}G^2\).

Covariance of conditional means. It remains to control \(\mathop{\mathrm{Cov}}(a_Pb_P,a_Qb_Q)\). The functions \(b_P,b_Q\) need not be monotone, but their changes are bounded by those of \(B_P,B_Q\). Indeed, (28) shows that \(B_P+b_P\) and \(B_P-b_P\) are nonnegative combinations of cosine means, multiplied by \(A_P\). Both functions are decreasing, by (23) and the decreasing rotator couplings. For ordered angle arrays \(\alpha\le\widetilde\alpha\), it follows that \[|b_P(\widetilde\alpha)-b_P(\alpha)| \le B_P(\alpha)-B_P(\widetilde\alpha).\] Set \(Y_P=a_Pb_P\) and \(D_P=a_P-B_P\). Combining this estimate with the increase of \(a_P\) and the bounds \(a_P,|b_P|\le1\) gives \[\begin{align*} |Y_P(\widetilde\alpha)-Y_P(\alpha)| &\le a_P(\widetilde\alpha)-a_P(\alpha) +B_P(\alpha)-B_P(\widetilde\alpha)\\ &=D_P(\widetilde\alpha)-D_P(\alpha). \end{align*}\] In particular \(D_P+Y_P\) and \(D_P-Y_P\) are increasing. Apply association to these functions and \(D_Q\pm Y_Q\). Adding the two inequalities with matching signs, and then the two with opposite signs, gives \[ |\mathop{\mathrm{Cov}}(Y_P,Y_Q)|\le\mathop{\mathrm{Cov}}(D_P,D_Q). \tag{29}\] Every term in the expansion \[\mathop{\mathrm{Cov}}(D_P,D_Q) =\mathop{\mathrm{Cov}}(a_P,a_Q)+\mathop{\mathrm{Cov}}(B_P,B_Q) -\mathop{\mathrm{Cov}}(a_P,B_Q)-\mathop{\mathrm{Cov}}(B_P,a_Q)\] is controlled by a single-component covariance already bounded above. For the two like-color terms, conditional covariances are nonnegative, so the covariances of conditional means are at most the full covariances. For the unlike-color terms, conditional independence identifies the covariances of conditional means with the full covariances exactly. All their lists are even sublists of the original two lists. Hence (29) bounds the second term of (27) by \(C_{r,s}G^2\) as well. Combining the two bounds proves (20). ◻

Proof. The finite-graph covariance bound passes to the infinite-time cylinder by transfer convergence. For sites on two rows separated by \(j\) time steps, their single-component correlation is at most \(\tau^j\): the vectors \(q_x^i\Omega\) are centered, have norm at most one, and \(T\) has norm \(\tau\) on \(\Omega^\perp\). Proposition 10 therefore gives, for every real even row polynomial \(P\) and its centered vector \(v_P=(P-\langle \Omega,P\Omega\rangle)\Omega\), \[ 0\le\langle v_P,T^jv_P\rangle\le C_P\tau^{2j}. \tag{32}\] Here we have summed the finitely many monomial covariance bounds; the constant need not be uniform in \(P,N\), or \(w\).

Polynomials in row components are uniformly dense in the continuous functions on the compact row space. Symmetrizing gives density of even polynomials in the even subspace. The continuous positive function \(\Omega\) has a positive minimum, so multiplication by \(\Omega\) is bounded and invertible. Hence the complex linear span of the vectors \(v_P\) is dense in the even vacuum complement. Since the spectral measure of each \(v_P\) is positive, (32) excludes any of its mass above \(\tau^2\): positive mass in \([\tau',1]\) with \(\tau'>\tau^2\) would give a lower bound proportional to \((\tau')^j\). Density proves (30).

Compactness of \(T\) implies that \(\tau\) is an eigenvalue. As \(\tau>\tau^2\), its eigenspace is odd. The three component reflections commute with each other and with the real operator \(T\), so one may choose a real unit eigenfunction with a definite parity under each reflection. Their product is simultaneous inversion, so at least one component parity is odd; relabel that component as \(3\).

The modulus \(|\psi|\) is even. Put \(b=\langle \Omega,|\psi|\rangle^2\). Positivity of the transfer kernel and (30) give \[\tau=\langle \psi,T\psi\rangle \le\langle |\psi|,T|\psi|\rangle \le b+(1-b)\tau^2.\] Rearranging yields \(b\ge\tau/(1+\tau)\), as claimed. ◻

Proof. Suppress \(N,w\) from the transfer notation. Let \(\psi\) be the real unit \(\tau\)-eigenfunction furnished in Section 4, where \(\tau=e^{-ae_N(w)}\), and choose color \(3\) so that \(\psi\) is odd under its sign flip. The row observable \(F=\mathop{\mathrm{sgn}}\psi\), with \(\mathop{\mathrm{sgn}}0=0\), is bounded by one and has the same oddness. Its torus correlation is \[\mathbb E_{\mathrm{tor}}[F(0)F(d)] =\frac{\mathop{\mathrm{Tr}}\bigl(F T^{d/a}F T^{(w-d)/a}\bigr)}{R_{N,w}(w)}.\] Every summand in its eigenbasis expansion is nonnegative. The summand with \(\psi\) on the segment of length \(d\) and \(\Omega\) on the other segment is \[\frac{e^{-de_N(w)}|\langle \Omega,F\psi\rangle|^2}{R_{N,w}(w)} =\frac{e^{-de_N(w)}\langle \Omega,|\psi|\rangle^2}{R_{N,w}(w)} \ge c\,e^{-(d+a)e_N(w)},\] by (31) and the uniform bound for \(R_{N,w}(w)\) in (15). Here \(\tau/(1+\tau)\ge\tau/2\); in particular, no bound on \(ae_N(w)\) is required.

If no cluster meets both rows, flip the signs of every cluster meeting the first row. Conditional on \(\alpha,z,\eta\), this preserves the law, changes the sign of \(F(0)\), and leaves \(F(d)\) unchanged. Thus the conditional expectation of their product is zero off \(\{0\longleftrightarrow d\}\), and its absolute value is at most one on that event. The claimed probability bound follows. ◻

Proof. By (36), the expectation of \(S(h_s,h'_s)\) is the corresponding normalized spin correlation. Translation invariance, plane moment convergence, and the strictly positive continuum kernel of Lemma 3 show that the cutoff plane expectation is bounded below by a positive constant for all large \(N\). Corollary 9 transfers this lower bound to all sufficiently large \(w\), uniformly in \(N,s\). Thus \(\mathbb E_{\mathrm{tor}}S(h_s,h'_s)\ge c_1>0\). Conditional Jensen and the uniform fourth moments give \[\mathbb E_{\mathrm{tor}}S(h_s,h'_s)^2 \le\mathbb E_{\mathrm{tor}}(X_{h_s}^2X_{h'_s}^2)\le C_1.\] The elementary second-moment inequality now yields \[\mathbb P_{\mathrm{tor}}\{S(h_s,h'_s)\ge c_1/2\} \ge\frac{c_1^2}{4C_1}.\] All constants already include the field normalization, so they are independent of the cutoff. ◻

Proof of Proposition 13. The union of the events \(D_{st}\) contains \(\{0\longleftrightarrow d\}\). Summing (39) over the \(O(w^2)\) pairs and applying (34) gives \[c\,e^{-(d+a)e_N(w)} \le\sum_{s,t}\mathbb E_{\mathrm{tor}}X_{H_s}X_{H_t}.\] Choose lifts with spatial center separation at most \(w/2\). The joint test supports then have spatial extent at most \(w/2+O(1)\) and time extent at most \(d+O(1)\), both less than \(3w/4\) for large \(w\). The uniform moment comparison therefore replaces each torus correlation by its cutoff plane value with error \(Ce^{-cw}\). For this fixed \(w\), there are only finitely many pairs of centers, so plane moment convergence applies to all of them as \(N\to\infty\). Their limiting correlations are bounded by \(Ce^{-m_*d}\): every pair of test supports has time separation at least \(d-O(1)\), and the kernel estimate (8) applies to the fixed test shapes. Consequently \[ \limsup_{N\to\infty}e^{-(d+a)e_N(w)} \le Cw^2\bigl(e^{-m_*d}+e^{-cw}\bigr). \tag{40}\]

Write \(L_w=\liminf_{N\to\infty}e_N(w)\). If \(L_w<\infty\), take a subsequence on which \(e_N(w)\to L_w\). Along it \(ae_N(w)\to0\), so (40) yields \[L_w\ge -\frac1d\log\!\left[ Cw^2\bigl(e^{-m_*d}+e^{-cw}\bigr)\right].\] If \(L_w=\infty\), the desired lower bound already holds. Since \(d=\lfloor\sqrt w\rfloor\), the displayed lower bound tends to \(m_*\) as \(w=w_k\to\infty\), proving (33). ◻

Proof. The threshold \(j_0/D\) differs from \(e^{-y}\) and \(e^{-(y+b_w)}\) by at least \(c b_w\) on their respective sides. This follows from the mean value theorem, uniformly on the fixed energy interval, since the rounding error is at most \(1/D=o(b_w)\). Hoeffding’s binomial tail bound (Hoeffding 1963, Theorem 1) therefore gives (44) and (45) without the factor \(e^{-t_0E}\), because \(Db_w^2=D^{3/4}\).

To retain a summable bound over all energies, that additional factor is essential. Choose \(c_1>0\) such that \(j_0\ge c_1D\) for every allowed \(y\) and all large \(w\). Fix \(E_*>4\log(2)/c_1+1\). For \(E\le E_*\), the loss \(e^{t_0E_*}\) is absorbed by decreasing the constant in the exponent \(-cD^{3/4}\), since \(t_0=O(D^{1/2})\). For \(E\ge E_*\), the binomial formula gives \[Q_{w,y}(e^{-E})\le 2^D e^{-j_0E}.\] For large \(w\), \(t_0\le c_1D/2\), and consequently \[2^D e^{-(j_0-t_0)E} \le \exp\!\left(-\frac{c_1}{4}DE\right) \le Ce^{-cD^{3/4}}.\] This proves (45) for every \(E\ge y+b_w\).

The smallest possible degree in the expanded polynomial is \(j_0\), which is at least \(t_0\) for large \(w\). Finally its coefficient sum in absolute value is bounded by \[\sum_{j=j_0}^D\binom Dj 2^{D-j}\le3^D.\] ◻

Proof of Proposition 16. Choose a fixed dyadic base width \(w_{k_0}\) large enough for the preceding estimates and so that \[\sum_{k=k_0}^{\infty}b_{w_k}<u_1-u_0.\] This is possible because \(b_{w_k}=O(w_k^{-1/16})\). The product of the factors \(1+Ce^{-c\lfloor\sqrt{w_k}\rfloor^{3/4}}\) over the same range is finite. At the base width, \[\mathcal N_{N,w_{k_0}}(u_1) \le e^{w_{k_0}u_1}R_{N,w_{k_0}}(w_{k_0})\le C_0,\] uniformly in the cutoff.

Fix a target \(w_k\) with \(k>k_0\). Apply (47) backwards along the finite chain \(w_k,w_{k-1},\ldots,w_{k_0}\), starting at threshold \(u_0\). The accumulated threshold shifts remain below \(u_1-u_0\), so every application is valid. For all sufficiently large \(N\) on this finite chain, we obtain \[\mathcal N_{N,w_k}(u_0) \le C_0\,2^{k-k_0} \prod_{h=k_0}^{k-1} \left(1+Ce^{-c\lfloor\sqrt{w_h}\rfloor^{3/4}}\right) \le Cw_k.\] Taking \(\limsup_{N\to\infty}\) proves the assertion. The cutoff threshold may depend on the target width; no simultaneous limit in \(N\) and \(w\) has been used. ◻

Proof. The spectral theorem and the character formula give \[\begin{align*} \int x^n\,\nu_{N,w,p}(dx) &=w\langle v,T_{N,w}^{n/a}\Pi_pv\rangle\\ &=a\sum_{\substack{b\in a\mathbb Z\\-w/2\le b<w/2}} e^{-ipb}\langle v,T_{N,w}^{n/a}U_bv\rangle. \end{align*}\] The matrix element in the sum is the cylinder correlation of the two bounded slab observables just described. For fixed \(f,n\) and large \(w\), their joint support has time extent at most \(n+2l\) and spatial extent at most \(w/2+\operatorname{diam}(\mathop{\mathrm{supp}}h)\). Both are below \(3w/4\). The cylinder-to-torus comparison, followed by Proposition 8, replaces this correlation by its cutoff plane counterpart with error at most \(Cw^2e^{-cw}\), uniformly in \(b\) and \(N\). The uniform field tails then remove the two truncations on the plane with error \(Ce^{-c'w}\), by Cauchy–Schwarz and the moment majorants. The factor \(a\) times the number of spatial translates is \(w\), so the total error in the displayed Riemann sum still tends to zero exponentially in \(w\).

For a fixed \(w\), the cutoff plane smeared correlations converge uniformly for \(|b|\le w/2\) to \(C_f(n,b)\). To see the uniformity, moment convergence gives pointwise convergence, while the difference-test bound (10) gives uniform equicontinuity. More explicitly, on a fixed compact translation interval the smooth translates satisfy \(|f_b-f_{b'}|\le C|b-b'|\) on a fixed finite union of unit boxes. The second-moment majorant bounds the \(L^2\) norm of the corresponding field difference by \(C'|b-b'|\). Cauchy–Schwarz then gives the same modulus of continuity for the correlations. Compactness upgrades pointwise convergence to uniform convergence. Consequently, as \(N\to\infty\), the Riemann sum of the cutoff plane correlations converges to the continuum integral over \([-w/2,w/2]\), uniformly for \(p\) in \(I\); the exponential factors have a uniform modulus of continuity on this compact set.

Finally, the separated-time kernel bound in Section 2 implies \(|C_f(n,b)|\le Ce^{-c|b|}\) outside a fixed compact interval. The omitted spatial tail therefore tends to zero uniformly in \(p\) as \(w\to\infty\). This proves (49). The case \(n=0\) is included: \(\mathop{\mathrm{supp}}g\) has positive distance from zero, so the two reflected field supports remain separated in time. ◻

Proof. Choose \(p_0>0\) so small that \[\sqrt{p_0^2+(5m_*/4)^2}<u_0, \qquad \widehat h(p)\ne0\quad (|p|\le p_0).\] This is possible because \(5m_*/4<u_0=3m_*/2\) and \(\widehat h(0)=\int h>0\). This momentum interval is fixed before any finite collection of masses is selected.

Choose \(J\) distinct masses \(\mu_1,\ldots,\mu_J\) from the indicated support. On \(I=[-p_0,p_0]\), the curves \[x_j(p)=e^{-E(p,\mu_j)},\qquad j=1,\ldots,J,\] are pairwise disjoint compact graphs in \(I\times(e^{-u_0},1)\). Their mutual distances and their distances from the two horizontal boundary lines are positive. There are therefore nonnegative continuous functions \(\Psi_j\) on \(I\times[0,1]\), supported in mutually disjoint neighborhoods of these graphs contained in \(I\times(e^{-u_0},1)\), and equal to one on the respective graphs.

For every \(p\in I\), \[L_j(p):=\int\Psi_j(p,x)\,\nu_p(dx)>0.\] Indeed, membership of \(\mu_j^2\) in \(\mathop{\mathrm{supp}}\rho\) gives positive measure to every neighborhood of that point. The integrand in (50) is positive there: \(\widehat h(p)\ne0\) and \(G(E)>0\) for every \(E>0\), since \(g\ge0\) is nonzero. Continuity of \(\Psi_j\) makes its value positive through a neighborhood of \(\mu_j\) at the fixed momentum. Moreover \(L_j\) is continuous in \(p\), by exponential domination in (50). Compactness yields \[\eta:=\min_{1\le j\le J}\min_{p\in I}L_j(p)>0.\] Both \(\eta\) and the required approximation scale may depend on this finite collection of masses.

By (51), for all sufficiently large fixed dyadic \(w\), and then all sufficiently large \(N\) depending on \(w\), \[\int\Psi_j(p,x)\,\nu_{N,w,p}(dx)>\eta/2 \qquad (j=1,\ldots,J)\] at every allowed \(p\in I\). Each atom of the measure (48) comes from a transfer-energy eigenspace with momentum \(p\). The disjoint supports of the \(\Psi_j\) therefore require at least \(J\) distinct energies below \(u_0\) in that space. Distinct momentum spaces are orthogonal. For large \(w\), the number of momenta in \(I\) is at least \(c_0w\), where \(c_0>0\) depends only on \(p_0\); at any fixed \(w\) these are distinct Brillouin characters once \(N\) is large. Thus \[Jc_0w\le\mathcal N_{N,w}(u_0).\] Taking the cutoff limit and using Proposition 16 gives \(Jc_0\le C\). The constants \(c_0,C\) are independent of the selected masses and of \(J\). The support can consequently contain only finitely many points in the stated interval. ◻

Proof of Theorem 1. The nonempty closed mass support has minimum \(m_*>0\), by the continuum inputs in Section 2. Proposition 19 makes this minimum isolated in the support. A positive locally finite measure assigns strictly positive finite mass to an isolated support point, so \[Z=\rho(\{m_*^2\})\in(0,\infty).\] Choose \(\delta>0\) small enough that no other mass in the support lies below \(m_*+\delta\), and define \[m_1=m_*,\qquad \rho_{\mathrm{rest}}=\rho-Z\delta_{m_*^2}.\] This is a positive measure, and its support is contained in \([(m_1+\delta)^2,\infty)\), as required. ◻