Methods and their relation to earlier work

The proof joins three traditions. Local rotation identities turn continuous symmetry into estimates on current fluctuations, as in the classical Ward-identity approach (Mermin 1967; Driessler et al. 1979; Aizenman and Simon 1980). Here the quaternion commutator supplies a quantitative decrease of stiffness across scales. The later passage from charged observables to invariant ones uses Ginibre’s formulation of the Griffiths inequalities for Ising and plane-rotator systems, the FKG association mechanism, and embedded reflection variables (Ginibre 1970; Fortuin et al. 1971; Wolff 1989), followed by two successive conditioning arguments. We prove the finite forms needed below and retain the covariances of conditional means at both stages.

Reflection positivity connects Euclidean lattice correlations with a positive transfer operator. Classical lattice reflection and chessboard methods were developed systematically by Fröhlich, Israel, Lieb and Simon (Fröhlich et al. 1978, 1980). The finite-volume argument uses the complete trace, so multiplicities and invariant sectors enter automatically. Its role differs from that of an infrared bound: it turns a small partition-function defect on one box into a spectral estimate uniform in larger circumferences. The positivity calculations and the limiting Hilbert-space argument are included in Section 2.

The block-variable viewpoint goes back to Kadanoff and Wilson (Kadanoff 1966; Wilson 1971). Constructive results for sigma models have required careful distinctions of regime and model. Kupiainen established a large-\(n\) expansion and a mass gap for sufficiently large \(n\) above the spherical model’s critical temperature (Kupiainen 1980); Gawędzki and Kupiainen constructed a finite-volume continuum limit for a hierarchical model (Gawędzki and Kupiainen 1986); Mitter and Ramadas constructed the Wilson trajectory in perturbation theory (Mitter and Ramadas 1989). Bałaban developed small-field renormalization and variational-background methods for vector spins (Bałaban 1995, 1996). For \(SU(2)\), identified with \(S^3\), Dybalski, Stottmeister and Tanimoto proved a small-field variational result for one block step, uniform in the box size (Dybalski et al. 2024b). These results are relevant predecessors of the block analysis; their hypotheses and conclusions do not supply the exact iterated \(O(4)\) density estimates required here.

Normalized stochastic block transformations for sigma models also appear in the perfect-action construction of Hasenfratz and Niedermayer (Hasenfratz and Niedermayer 1994, sec. 2.1, Eqs. (3)–(5)). The observation kernel used here differs, and its iterated estimates are proved below. The renormalization step is an exact integration through that probability kernel. Its estimates concern the complete density, including rough configurations, rather than only a formal expansion near aligned spins. Perturbative coefficients determine the scale only after these uniform estimates are available. The comparison of two cutoffs then requires equal terminal kinetic couplings. Since the retained expansion has signed terms, we compare complete positive densities on a common set of large probability under both laws. These are the two interfaces that connect the coefficient calculation with the nonperturbative gap.

Proof strategy

There are two distinct changes of scale in the proof. At fixed \(\beta\), we enlarge a periodic box to control the infinite-volume spectrum. To compare ultraviolet cutoffs, we instead change the bare coupling and the number of blocking steps while holding the coarse coupling fixed. The argument succeeds because the second operation preserves a finite-box estimate to which the first operation applies.

Transfer eigenvalues and a measure of excited-state weight

Fix a spatial circumference \(w\). A time slice is a configuration \(s=(s_1,\ldots,s_w)\in(S^3)^w\). Set \(V_w(s)=\sum_{i=1}^w s_i\cdot s_{i+1}\), with periodic indices, and let \(K_w\) be the integral operator with kernel \[ K_w(s,s')= \exp\!\left[\frac\beta2 V_w(s)+\beta\sum_{i=1}^w s_i\cdot s'_i +\frac\beta2V_w(s')\right]. \tag{12}\] The operator is compact and self-adjoint. It is positive semidefinite: the expansion of \(e^{\beta s\cdot s'}\) is a sum of positive tensor-product kernels, and multiplication on both sides by \(e^{\beta V_w/2}\) preserves positivity. The positive-kernel series has finite diagonal integral, so it also proves trace class. The strictly positive kernel gives a simple largest eigenvalue. Write \(\lambda_1(w)>\lambda_2(w)\ge\lambda_3(w)\ge\cdots\ge0\), allowing \(\lambda_2=0\). Closing the time direction gives \[ Z_\beta(n,w)=\operatorname{Tr}K_w^n=\sum_i\lambda_i(w)^n. \tag{13}\]

Suppress \(\beta\) temporarily and define \[\begin{align*} p(n,w)&=2\log Z(n,w)-\log Z(2n,w),\tag{14}\\ q(n,w)&=2\log Z(n,w)-\log Z(n,2w). \tag{15}\end{align*}\] Both are nonnegative. To interpret \(p\), introduce the normalized trace weights \[ t_i=\frac{\lambda_i(w)^n}{\sum_j\lambda_j(w)^n}. \qquad\text{Then}\qquad e^{-p(n,w)}=\sum_i t_i^2. \tag{16}\] Thus small \(p\) means that almost all the trace weight lies in one eigenstate, necessarily the ground state. The same interpretation applies to \(q\) after interchanging the coordinate directions.

Increasing the spatial volume

Ground-state dominance at one fixed circumference is not enough. New low-energy states could appear as the circumference increases. The following exact identity lets us control that limit: \[ p(n,2w)=2p(n,w)-2q(n,w)+q(2n,w). \tag{18}\] It follows by substituting (14)–(15). There is a second identity obtained by interchanging the directions.

Figure 1 records the two-direction geometry.

Define the square-box test \[ \Delta_\beta(n)=4\log Z_\beta(n,n)-\log Z_\beta(2n,2n). \tag{19}\] It obeys \(\Delta_\beta(n)=2p(n,n)+q(2n,n)\ge0\). The coefficients in (19) cancel any term proportional to the area of the box.

The slab estimate extends to bounded measurable local observables by approximation in their finite-dimensional distributions. No prefactor depending on the size of a slab is needed; this will be useful for the upper mass bound.

The criterion has reduced the problem to a single question: can one make \(\Delta_\beta(n)\) small with \(n\) no larger than a fixed multiple of \(\rho(\beta)^{-1}\)? The preliminary estimate alone will not do this. It will instead supply a reference box for the comparison between cutoffs.

Local class and Ward identities

The following calculation is a local Ward identity, in the tradition of (Mermin 1967; Driessler et al. 1979; Aizenman and Simon 1980). We give the full quaternion formula because the noncommuting current term and its factor two determine the quantitative stiffness gain.

Work on \(Q(S)=\{0,\ldots,S\}^2\) with all boundary spins fixed. For each positively oriented edge \(e=x\to y\), set

\[t_e+s_e=q_yU_eq_x^{-1},\qquad t_e\in\mathbb R,\quad s_e\in\operatorname{Im}\mathbb H\simeq\mathbb R^3,\]

where \(U_e\) is any fixed unit quaternion. Take signed coefficients \(|b_e|\leq b\leq B\), with \(w_e=b_e\) except that edges tangent to the boundary carry \(w_e=b_e/2\). The local density is \(Z^{-1}\exp(\sum_e w_et_e)\). The half allocation makes adjacent cell actions sum exactly to the action on the union.

For macroscopic midpoint coordinates \(z_e\), define

\[X_Q(F)=S^{-1}\sum_e w_es_e\cdot F_{\mu(e)}(z_e),\]

\[ \mathcal H_Q(F,G)=S^{-2}\mathbb E\sum_e w_et_e\,\overline{F_e}\cdot G_e -\operatorname{Cov}(X_Q(F),X_Q(G)). \tag{28}\]

The covariance is sesquilinear, with conjugation on the first entry. Write \(H_Q\) for the restriction to constant \(3\times2\) matrices. In each spatial direction the sum of the absolute edge coefficients is at most \(bS^2\): there are \(S(S-1)\) interior edges and two tangent boundary rows with half weight. Thus

\[ H_Q\leq bI. \tag{29}\]

This is valid with signed coefficients and twists. No positive lower bound on \(H_Q\) is claimed or used.

For an imaginary unit \(a\), rotate \(q_x\mapsto e^{af_x}q_x\). A bond quaternion changes to \(e^{af_y}(t_e+s_e)e^{-af_x}\). Its \((1,a)\)-plane rotates through \(f_y-f_x\); its imaginary plane perpendicular to \(a\) rotates through \(f_y+f_x\). If \(\mathcal A=-\sum w_et_e\), then

\[ D_f\mathcal A=\sum_e w_e(s_e\cdot a)(f_y-f_x),\qquad D_f^2\mathcal A=\sum_e w_et_e(f_y-f_x)^2. \tag{30}\]

For a general infinitesimal imaginary profile \(\eta_x\), the imaginary bond derivative is exactly

\[ D_\eta s_e=t_e(\eta_y-\eta_x)+(\eta_y+\eta_x)\times s_e. \tag{31}\]

Consequently the compact-profile Ward identity has a contact term and a bracket term. If \(\eta\) vanishes on the fixed boundary, Haar integration by parts gives

\[ \mathbb E[\overline{X_Q(d_S\eta)}X_Q(F)] =S^{-2}\mathbb E\sum_e w_et_e\overline{(d_S\eta)_e}\cdot F_e +\mathbb E X_Q\!\left(F_e\times(\overline{\eta_y}+\overline{\eta_x})\right), \tag{32}\]

where the cross product is extended complex bilinearly. This follows first for real profiles and then by complex linear extension. Here \((d_S\eta)_\mu(z)=S[\eta(z+e_\mu/(2S))-\eta(z-e_\mu/(2S))]\). This fixes the normalization and the factor two in the eventual nonabelian derivative. For the same-axis score there is no bracket term; (30) follows directly.

Current moments near a pinned boundary

Assume throughout the local induction

\[ \log(2+S)\leq16B. \tag{33}\]

Conditioning into cells

The symbol \(L\) in this preliminary argument is an auxiliary integer which may change during the stiffness iteration. It is independent of the fixed block size chosen for the exact renormalization map in Section 4.

The current-moment estimate permits replacing slowly varying fields by their values at cell anchors. We next express the large-square stiffness through the conditioned cell laws; the variance of their conditional currents is the gain to exploit.

Tile \(Q(S)\) by \(L^2\) cells of side \(R=S/L\). Condition on their complete boundary rings \(\omega\). Cell interiors are independent and the ring density has action \(S_g(\omega)=-\sum_c\log Z_c(\omega)\). Set

\[j(F)=\mathbb E[X_Q(F)\mid\omega] =L^{-1}\sum_c\mathbb E_cX_c(F).\]

Let \(z_c\) be the lower-left cell anchor and put

\[N_c(F)=\sup_{c^+}(|F|+L^{-1}|\nabla F|),\qquad D_c(F)=L^{-1}N_c(\nabla F).\]

The fixed enlargement \(c^+\) is used only for smooth test-field estimates. Write \(P(B)\) for a fixed polynomial whose degree and coefficients may depend on fixed moment orders or prescribed exponents, but not on \(R,L,S\). Equation (34) and (33) give

\[ |\mathbb E_cX_c(F)|\leq P(B)N_c(F),\quad |\mathcal H_c(F,G)|\leq P(B)N_c(F)N_c(G), \tag{37}\]

\[ |\mathcal H_c(F,G)-\overline{F_c}H_cG_c| \leq P(B)[D_c(F)N_c(G)+N_c(F)D_c(G)]. \tag{38}\]

For the last estimate subtract the constants \(F_c,G_c\) in the bilinear form. Their remainders have cell-coordinate \(C^1\) norm bounded by the displayed \(D_c\). One can instead use a direct Lipschitz bound on the cell where appropriate.

For a scalar ring rotation \(u\), let \(H_c^u\) be the resulting constant-field stiffness and \(O_c(u)=\operatorname{Ad}(e^{au(z_c)})\). Then

\[ \|O_c(u)^TH_c^uO_c(u)-H_c\|\leq P(B)D_c(u), \tag{39}\]

and its derivative at zero satisfies the same bound. To prove this, remove the constant rotation \(u(z_c)\) exactly by covariance. Along the remaining path also rotate the interior integration variables. The logarithmic density derivative is the negative of the centered cell current with test \(aR(u_y-u_x)\), whose fixed moments are bounded by \(P(B)D_c(u)\). The residual profile need not vanish on the cell boundary; the current-moment bound applies to this test without that restriction. Derivatives of a constant-field current have the bracket current with residual profile of size \(D_c(u)\), plus a deterministic contact of that size. Differentiate the contact and covariance in (28) and apply Hölder with the already established current moments. Restarting this calculation at each point of the path gives the bound for arbitrary amplitude \(u\); there is no exponential dependence on its constant part.

Rotating all rings while fixing the outside boundary gives the exact Hessian identity

\[ D_u^2S_g=L^{-2}\sum_c\mathcal H_c(a\,d_Su,a\,d_Su). \tag{40}\]

Gaussian augmentation and commuting derivatives

The cell stiffness matrices need not be positive. We add auxiliary Gaussian noise so that the desired upper bound on \(H_Q\) becomes a lower bound on the variance of an augmented current. The identity below makes this reduction exact.

Suppose uniformly throughout the pinned cell class \(H_c\leq hI\), where \(B^{-D}\leq h\leq B\). Choose

\[ h<v\leq2B,\qquad c_0\geq B^{-D'},\qquad C_c=vI-H_c-c_0P_a\geq B^{-D'}I, \tag{41}\]

where \(P_a\) is projection onto the two spatial components of color \(a\). Let \(n_c\) be independent standard six-dimensional Gaussian vectors, and \(n_{0,c}\) independent two-dimensional Gaussian vectors of covariance \(c_0I\), all independent of the true ring data. Define

\[ J(F)=j(F)+L^{-1}\sum_c(\sqrt{C_c}n_c+a n_{0,c})\cdot F_c. \tag{42}\]

For every constant real matrix \(A\), total variance gives exactly

\[ H_Q(A,A)=v|A|^2-\operatorname{Var}J(A). \tag{43}\]

Indeed \(H_Q=L^{-2}\sum_c\mathbb EH_c-\operatorname{Var}j\), and the conditional noise covariance adds \(vI-L^{-2}\sum_cH_c\).

Choose a real trigonometric cutoff \(\psi\), zero on the boundary, with band \(O(m)\). Let \(f_i=\psi\varphi_i\), where \(\varphi_i\) are the real orthonormal Fourier modes of frequency \(|k|\leq4K\), including the constant. Use \(K+m\ll L\) and

\[G_0=L^{-2}\sum_c\nabla f(z_c)\nabla f(z_c)^T, \quad p=(c_0G_0)^{-1}L^{-1}\sum_c\nabla f(z_c)n_{0,c}, \quad \theta=f\cdot p.\]

Quadrature is exact at this band, so \(G_0\) is the continuum gradient Gram matrix. It is positive: a linear combination of the \(f_i\) with zero gradient is a constant, zero on the boundary, and hence is zero; division by \(\psi\) on its open nonzero set then annihilates the Fourier polynomial. Thus \(p\) is Gaussian with covariance \((c_0G_0)^{-1}\).

Put

\[\omega^0=e^{-a\theta}\omega,\qquad n_{0,c}^0=n_{0,c}-c_0\nabla\theta(z_c)/L,\]

\[n_c^0=O_c(\theta)^Tn_c-\sqrt{C_c^0}\,a\nabla\theta(z_c)/L,\]

where \(C_c^0\) uses \(\omega^0\). At fixed base data \((\omega^0,n_0^0,n^0)\), the density of \(p\) is proportional to \(e^{-V(p)}\), with

\[ V(p)=\tfrac12c_0p^TG_0p+S_g(e^{a\theta}\omega^0) +\tfrac12\sum_c|n_c^0+\sqrt{C_c^0}\,a\nabla\theta(z_c)/L|^2. \tag{44}\]

This is an exact change of variables. First split the Gaussian \(n_0\) into its linear projection and orthogonal residual. At fixed \(p\), the ring rotation preserves product Haar measure. At fixed \(p,\omega^0\), the \(n_c\) transformations are orthogonal maps followed by translations. These successive maps have constant Jacobian; dependencies of later maps on earlier variables give a triangular Jacobian, not an extra determinant.

For fixed scalar directions \(u_i\) in the potential span, let \(\partial_i\) mean addition of \(tu_i\) to \(\theta\) with all base data fixed. They commute. Conditional integration by parts gives, for \(\partial_i^*=-\partial_i+\partial_iV\),

\[ \mathbb E\left(\sum_i\partial_i^*Y_i\right)^2 =\mathbb E\sum_{ij}(\partial_jY_i)(\partial_iY_j) +\mathbb E\sum_{ij}Y_i(\partial_i\partial_jV)Y_j. \tag{45}\]

The identity first holds conditionally and then in the full law. The conditional density has Gaussian decay: its first term is strictly positive quadratic and \(S_g\) is bounded for each fixed finite system. The functions involved and their derivatives have at most polynomial growth in Gaussian variables, since the matrices remain in a fixed positive cone for that system. Thus the integration by parts has no missing boundary term.

Three uniform band estimates

The integration-by-parts identity in the preceding subsection contains a Hessian term and a crossed derivative term. We next control their ingredients uniformly in the number of modes, while retaining the exact contact cancellation in the Ward identity.

Fix \(D,D'\) and any requested power \(N\). Take \(K/L\leq B^{-J}\), with \(J\) sufficiently large depending only on these fixed numbers and on the fixed test amplitude bounds. Retain \(S\geq L\), (33), and \(Bm\leq K\). The constants denoted \(P(B)\) below do not depend on the subsequently increased \(J\).

A weighted current estimate

The band estimates control differentiated currents. We now sum them with frequency weights whose convolutions remain summable. This is the auxiliary estimate needed to bound the crossed derivative term in the variance gain; it is not yet the stiffness improvement itself.

Fix an integer \(N_*>100(1+D+D')\) and apply the band estimates with accuracy \(B^{-10N_*}\). Increase the fixed exponent \(J\) whenever required below. The error estimates in the next two subsections are then bounded by \(C B^{-N_*}\). To track this convention, all remaining weighted frequency sums are bounded by fixed powers of \(T,T_w\leq CB\); all cutoff derivative and cell-moment constants are fixed powers of \(B\). The only crude factors in \(L\) occur with an \(S^{-2}\) Taylor remainder or an event of probability \(e^{-B^3}\). Since \(S\geq L\) and \(\log L\leq16B\), those errors are also at most \(B^{-N_*}\). No unweighted count of the main frequency band is used in this convention.

Choose a sufficiently large fixed exponent \(J\) and set

\[M=B^J,\quad K=L/B^J,\quad L\geq B^{10J}, \quad \psi(z)=\prod_{\mu=1}^2[1-\cos^{2m}(\pi z_\mu)], \quad m=\lceil B^4\rceil.\]

All fixed powers of \(\psi\) have Fourier \(\ell^1\) norm bounded independently of \(B\), band \(O(m)\), and fixed derivative bounds polynomial in \(B\). Put \(\chi=\psi^2\); then

\[ I_\chi=\int_{[0,1]^2}\chi^2\geq1-CB^{-2}. \tag{60}\]

This follows from the Gaussian decay of \(\cos^{2m}(\pi z)\) away from the seams and an exceptional width \(O(m^{-1/2})\).

In a long step use \(K_1=L/B^{4J}\), \(K_2=L/B^{3J}\). In a short step use \(K_1=8M\), \(K_2=256M\). The main reciprocal lattice is \(2\pi\mathbb Z^2\), and

\[ T=\sum_{M\leq|k|\leq K_1}|k|^{-2} =\frac1{2\pi}\log(K_1/M)+O(M^{-1}),\qquad c\leq T\leq CB. \tag{61}\]

The coefficient is \((2\pi)^{-2}\times2\pi=1/(2\pi)\). The sum here includes both signs of the Fourier frequencies, equivalently both normalized real modes for each pair.

The variance gain

We now use the weighted estimate to obtain a lower bound on current variance. Through (43), this becomes the desired decrease of the stiffness upper bound.

For a real constant \(3\times2\) matrix \(A\) of Hilbert–Schmidt norm one, choose a unit \(a\) perpendicular to both its columns. This is where the three-dimensional imaginary quaternion space is used. Put \(A'=-a\times A\), so \(|A'|=1\). On the main annulus set

\[u_i=\chi\varphi_i/|k_i|,\quad G_i^{A'}=d_S\bigl(\chi(A'\widehat k_i)\widetilde\varphi_i/|k_i|\bigr),\quad Y_i=J(G_i^{A'})/|k_i|.\]

The gradient Gram operator of the \(u_i\) is bounded by \((1+Cm/M)^2\). The Ward bound, exact quadrature, and \(\sum|k|^{-2}\widehat k\widehat k^T=(T/2)I_2\) yield

\[ \sum_i\mathbb E Y_i^2=vI_\chi T/2+O(B^{-N_*}),\qquad \sum_i\mathbb E(\partial_iJ(A))Y_i=vI_\chi T+O(B^{-N_*}), \tag{67}\]

\[ \mathbb E Y^TV''Y\leq v^2I_\chi T/2+O(B^{-N_*}). \tag{68}\]

The second formula in (67) uses the factor two in (52). The leading-symbol replacement of \(G_i\) is made only inside the contact pairing after applying (59), with error \(C[m/|k_i|+(|k_i|/S)^2]\). Summing with \(|k_i|^{-2}\) makes the total arbitrarily small. The same error convention, chosen stronger than all needed powers of \(v\), is used in this section.

Iteration of the stiffness bound

Fix any sufficiently large \(D\), then \(D'>D\), and fix \(J\) to make all preceding errors as small as required. Start with the uniform upper bound \(H_1\leq bI\). If \(b\leq B^{-D}\), no step is needed.

For \(h\geq A\log B\), use

\[v=h+1,\quad c_0=1/2,\quad L=\left\lceil e^{\sqrt{h\log B}}\right\rceil.\]

Choose fixed \(A\) large compared with \(J^2\) and the constants in (75). Then \(L\geq B^{10J}\), \(\log L\leq h\), (41) holds, and (43), (75) decrease the stiffness upper bound by

\[\delta h\geq(\log L)/\pi-C\log B \geq c\sqrt{h\log B}.\]

There are at most \(C\sqrt{B/\log B}\) such steps. Summing the relation \(\log L\leq\pi\delta h+C\log B\) gives a total logarithmic side length at most

\[ \pi b+C_D\sqrt{B\log B}. \tag{77}\]

Stop at \(h\leq C\log B\), using a weaker positive upper bound at the crossing if necessary. Now take a small fixed \(\delta<c_{\rm sh}/4\), and use

\[v=h+\delta\min(h,1),\quad c_0=\tfrac12\delta\min(h,1),\quad L=\lceil B^{11J}\rceil.\]

The matrix lower bounds in (41) hold with \(D'>D\), after increasing the fixed lower threshold for \(B\). Equations (43), (76) reduce \(h\) by at least \(c\min(h,1)\). It takes \(O_D(\log B)\) steps to reach \(B^{-D}\), at total additional logarithmic length \(O_D((\log B)^2)\). This is absorbed in (77).

At all stages \(\log(2+S)\leq16B\) holds once \(B\) exceeds a threshold depending only on the fixed choices: the accumulated bound is \(\pi b+o(B)\), and a proposed next step adds at most \(B\). This verifies the premise used in the moment bounds, rather than assuming it retrospectively.

We have proved, uniformly over the signed/twisted/pinned class, that there is a common integer side \(R_*\) with

\[ \boxed{\ H_{R_*}\leq B^{-D}I,\qquad \log R_*\leq\pi b+C_D\sqrt{B\log B}.\ } \tag{78}\]

Uniform control of local rotations

Take coarse cells of integer side comparable to \(s=R_*M_*\), where \(M_*=\lceil B^{D+C_0}\rceil\). Rectangles with sides in \([s,2s]\) allow arbitrary sufficiently large tori to be tiled. Profiles below are supported on a fixed number of coarse cells, vanish at the outer boundary of a fixed collar, are continuous across cells, and have uniformly bounded cellwise \(C^2\) extensions. Pins lie outside the collar.

For one fixed-axis profile \(f\), the effective coarse ring action \(F\) obeys

\[ D_f^2F\leq CB^{-D} \tag{79}\]

at every ring configuration. To prove it, subdivide each coarse cell into side-\(R_*\) squares and possible leftover strips of total area \(O(sR_*)\). Conditional variance subtraction bounds its Hessian above by the sum of the subcell Hessians. The gradient test on a small square has size \(O(M_*^{-1})\), with variation smaller by another \(M_*^{-1}\). Equations (78), (38) bound each small-square cost by \(CM_*^{-2}[B^{-D}+P(B)/M_*]\). Summing gives \(CB^{-D}+P(B)/M_*\). On leftover strips discard the negative variance and use the contact bound: their total cost is \(CBR_*/s=CB/M_*\). Choose \(C_0\) large enough. The polynomial in this use of (38) is independent of \(D\), so the choice is not circular.

Haar integration by parts on the free ring spins gives

\[ \mathbb E(D_fF)^2=\mathbb ED_f^2F\leq CB^{-D}. \tag{80}\]

For the normalized square-root density \(\varphi\) relative to product Haar, this is \(\|D_f\varphi\|_2^2\leq CB^{-D}\). Pullbacks by deterministic ring rotations are unitary. Integrate the derivative along a fixed-axis rotation and telescope a bounded number of such factors. For every allowed product \(\phi\),

\[ \mathbb E\left(e^{-[F(\phi\omega)-F(\omega)]/2}-1\right)^2\leq CB^{-D}. \tag{81}\]

Thus a specified action displacement exceeds fixed \(\epsilon>0\) with probability \(O_\epsilon(B^{-D})\).

Coarse orientations and decay of charged observables

Independently adjoin Haar variables \(g_X\in S^3\) at free coarse vertices, equal to identity at pinned boundary vertices. On a directed coarse edge \(X\to Y\), interpolate by

\[\Phi(g;t)=g_X\exp[\eta(t)\ell(g_X^{-1}g_Y)],\]

where \(\eta\) is fixed smooth and constant near endpoints, and \(\ell\) is any fixed measurable logarithm of norm at most \(\pi\), equal to zero at identity. Write the retained spin ring as \(\omega=\Phi(g)\widetilde\omega\). At fixed \(g\) this is a Haar-preserving change of ring variables. Conditional on \(\widetilde\omega\), the \(g\)’s have density proportional to \(e^{-F(\Phi(g)\widetilde\omega)}\), a finite-range Gibbs density.

For every unpinned cell its potential is invariant under common left multiplication of its incident \(g\)’s: \((kg_X)^{-1}(kg_Y)=g_X^{-1}g_Y\), so \(\Phi(kg)=k\Phi(g)\), and simultaneous left rotation of the cell’s rings and interior preserves its action even with the fixed right twists.

Call an interior vertex bad if, for some values of the neighboring \(g\)’s, varying its own \(g_X\) changes the incident-cell action by more than \(\epsilon\). Enlarge the test to permit all logarithms of the prescribed bounded size satisfying the endpoint equations, using the same data on unaffected edges. This enlargement makes the test a supremum over a compact parameter set and covers every branch choice of the measurable interpolation.

Mixing for arbitrary local observables

From now on use the untwisted zero-field ferromagnet, with \(0\leq b_e\leq b\). This restriction is essential. Neither the association argument nor its conclusion is being claimed for arbitrary pinned or frustrated systems.

Periodic limits and the partition-function bound

Fix the absolute \(D\) required by the profile parameter dimension, then all other constants, and take \(B=\max(B_*,b)\). The polynomial overhead in \(s=R_*M_*\) is absorbed by (78), so one may use

\[ s(b)\leq\exp[\pi b+C\sqrt{B\log B}]. \tag{92}\]

The free precision and the conditional mean

Let \(h\) count block transformations from the nearest-neighbor precision. Define \(K_0=D\) and \[ K_{h+1}(p)^{-1}=a^{-1} +L^{-2}\sum_l|b(k_l)|^2K_h(k_l)^{-1}. \tag{95}\] The pole at zero in this expression is interpreted by continuation. Equation (95) is the covariance of \(Qu\) plus independent Gaussian observation noise.

The conditional mean also gives an exact energy decomposition. For real ambient fields \(u,V\) on a finite periodic pair of lattices, \[ \frac12\langle u,K_hu\rangle+\frac a2\|V-Qu\|^2 =\frac12\langle V,K_{h+1}V\rangle +\frac12\langle u-P_hV,C_h^{-1}(u-P_hV)\rangle. \tag{100}\] Indeed \(C_h^{-1}P_h=aQ^*\) and \(a(I-QP_h)=K_{h+1}\) make the cross and constant terms agree after expansion. Although \(K_h\) vanishes on constants, \(C_h^{-1}=K_h+aQ^*Q\) is strictly positive, since \(Q\) preserves constants. This identity identifies the fluctuation field and its positive precision. The next construction expresses the energy through localized edge rows, with the row and column estimates needed to keep that positivity near rough regions.

A positive identity for the kinetic energy

The edge representation of \(K_h\) is not unique: a curl correction leaves \(d^*B_hd\) unchanged. We use this freedom to obtain a positive completion whose fine-output rows gain \(L^{-1}\). All operators act on real ambient fields, so the resulting identity also applies to sphere-valued fields without linearizing them.

Scales, observation, and the class of densities

Choose a dyadic integer \(L\), later sufficiently large, and put \[ \begin{aligned} \gamma&=\frac{\log L}{\pi},& H_j&=H+\gamma j,& \mathfrak m_j&=\left\lceil(\log H_j)^2\right\rceil,\\ p_j&=(\log H_j)^{P_0},& t_j&=H_j^{-1/2}p_j,& T_j&=H_j^{-.49}. \end{aligned} \tag{106}\] The exponent \(P_0\) and then \(H\) will be fixed below. A run of depth \(N\) uses layers \(j=N,N-1,\ldots,0\) and free history \(h=N-j\). At a step its precise coupling \(b\) is required to lie in \([H_j/2,2H_j]\). We write \(g=b^{-1/2}\) and omit \(j,h\) temporarily. Within a single step, \(M=\mathfrak m_j\) and \(M'=\mathfrak m_{j-1}\). Thus \(t\asymp gp\). All geometric thresholds in a comparison use \(H_j\), not the varying \(b\).

On a square torus of side \(mL^j\), take \(m\) dyadic and \(m\ge C_{\rm adm}\mathfrak m_0\). The constant \(C_{\rm adm}\) is fixed sufficiently large for the finite neighborhoods below. Increasing \(H\) gives \(\mathfrak m_j/\mathfrak m_{j-1}\le2\); hence these period conditions hold simultaneously at every layer. Bulk formulas mean the same formulas with their absolutely summable position lists on \(\mathbb Z^2\).

The observation from \(q\) to one new spin \(V_Y\) in each block is \[ \begin{aligned} \mathcal K_b(dV\mid q)&= \mathbb E_{\tau}\!\left[ \prod_Y\frac{\exp\{-ba|V_Y-\widehat m_Y(q,\tau_Y)|^2/2\}} {z(ba)}\,dV_Y\right],\\ z(ba)&=\int_{S^3}e^{-ba|V-1|^2/2}\,dV, \qquad m_Y=Qq(Y). \end{aligned} \tag{107}\] Here \(a\) is the fixed positive observation precision in the free construction. If \(|m_Y|\ge1/2\), set \(\widehat m_Y=m_Y/|m_Y|\). Otherwise the independent tag \(\tau_Y\) samples a constituent spin with the weights of \(Q\), and \(\widehat m_Y(q,\tau_Y)\) is that spin. The expectation in (107) averages these tags. They may be included at every block, even where they are unused. The denominator in (107) is independent of the unit vector \(\widehat m_Y\), so this is a probability kernel.

For an edge \(e\), write \(r_e=|d_eq|\). Fix \(0<d_0<c_0/16\) and a constant \(R_*\) sufficiently large compared with the exponential localization length of \(\ell_h\). Define \[\begin{align*} S_t(r)&=\begin{cases}c_0r^2/2,&r\le t,\\d_0tr,&r>t,\end{cases} \\ m_e(q)&=\mathbf1\{r_f\le t e^{\operatorname{dist}(e,f)/R_*} \text{ for all }f\},\\ \mathcal E_{h,t}(q)&=\sum_eS_t(r_e) +\frac12\sum_em_e(q)| (\ell_hdq)_e|^2. \tag{108}\end{align*}\] The mask reads only edges within \(O(\log(1/t))\) of \(e\), since chord lengths are at most two. Moreover \(m_e=1\) implies \(| (\ell_hdq)_e|\le Ct\). The masks and \(S_t\) may be discontinuous; they will never be differentiated as functions of a moving output configuration.

We next specify the locality norms. Each label carries its complete support and a connected record of cubes of radius \(\mathfrak m_j\) covering that support, together with the paths used to join them. Its load is an integer \(s\ge1\). The conventions are chosen so that the support has at most \(C\mathfrak m_j^2s\) sites and its record has length at most \(C\mathfrak m_js\). Every argument, mask, eligibility test, negative status query, and window determining a formula is included. On a torus the record retains the lifted displacements before folding. Intersecting records can be joined with load at most a fixed multiple of their total load.

Projection of a record to the next lattice, followed by any of the fixed neighborhoods used below, has a record satisfying \[ 4s/L\le s'\le D_*(1+s/L). \tag{109}\] The upper bound follows by breaking its walk into pieces of length comparable with \(L\mathfrak m_{j-1}\); the lower bound is imposed by retaining unused load if necessary. A record of load less than \(c_*mL^j/\mathfrak m_j\) is called short. Choose \(c_*\) small enough that every short complete record has an injective lift. Its formula must coincide with that of the corresponding bulk label. All other records are called winding records, whether or not their displayed support itself wraps. This convention also applies to a constant or a short function whose derivation used a long record.

For a connected graph \(X\) let \(1_X(q)=\mathbf1\{r_e\le t_j\text{ for all }e\in X\}\). At an anchor \(o\), use relative coordinates \[y_o(x)=\operatorname{Im}(q_xq_o^{-1}),\qquad S_o(q)=\frac14\sum_{|e|=1}|y_o(o+e)|^2.\] Fix a basis of each space of invariant tensors \(((\mathbb R^3)^{\otimes n})^{SO(3)}\), \(2\le n\le6\). In particular this includes the alternating cubic tensor. A canonical polynomial is a homogeneous tensor polynomial in the \(y_o(x)\), with an anchored joining path of length \(u\) comparable with \(1+\sum_x|x-o|\), counted with multiplicity. Its coefficient norm is \[ \|A_n\|_\sigma= \sup_o\sum_{\text{labels at }o}e^{\sigma u}(1+u)^2 |a_{\text{label}}|. \tag{110}\] Take \(\sigma>0\) sufficiently small for the finitely many uses of the free exponential bounds; it is fixed before \(L\). The seven canonical slots, in their order, are \[ bP_4,\quad I_2,\quad bP_5,\quad I_3,\quad bP_6,\quad I_4,\quad b^{-1}J_2. \tag{111}\] The subscript records homogeneous field degree, and \(\mathbf P=(P_4,I_2,P_5,I_3,P_6,I_4,J_2)\) denotes the seven coefficient lists without their displayed powers of \(b\). Their norms have fixed caps \(B_1,\ldots,B_7\), chosen successively below.

Each quadratic label is grouped into its dihedral orbit at the anchor and compensated by a multiple of \(S_o\), so that its Hessian vanishes on unit affine fields at the identity. This means its second derivative on \(q_x=\exp\{\theta v\,\widehat e\cdot(x-o)\}\) is zero, for a unit imaginary quaternion \(v\) and a coordinate unit vector \(\widehat e\). Spin and spatial symmetry then imply vanishing on every pair of affine tangent fields. Compensation coefficients are included in (110). All polynomials are formed before folding. For \(u\le \mathfrak m_j\) their mask graph may be a ball of radius \(C_\chi \mathfrak m_j\) at \(o\). Longer paths are padded by fixed multiples of \(\mathfrak m_j\). Their load is at most \(C(1+u/\mathfrak m_j)\).

A regular error \(f_X\) is defined and \(C^k\), with \(k=8\), on \(\{r_e<4t_j:e\in X\}\). Its norm \([f_X]_{k,j}\) is the sum of the sup norms of its derivatives through order \(k\), using independently in each derivative slot the bounds \[ |v_x|\le t_j(1+\operatorname{dist}(x,o))^8. \tag{112}\] Derivatives are taken in \(q_x(\theta)=\exp(\sum_i\theta_iv_{i,x})q_x\). On a torus use physical distances. The error norm and cap are \[ \|f\|_{k,j,A}:=\sup_o\sum_{X\text{ at }o}e^{As_X}[f_X]_{k,j} \le\delta_j:=H_j^{-2.05},\qquad A=64. \tag{113}\] Each bulk or nonwinding error has zero value and zero affine Hessian at the identity. Its first derivative is zero by conjugation invariance. Winding errors need not have these normalizations.

Finally a covering label \(\ell\) has a bounded measurable weight \(k_\ell(q)\), complete support \(P_\ell\), and a nonempty inventory \(D_\ell\) of bonds in that support. Its weight vanishes unless \(D_\ell\subset D(q):=\{e:r_e>t_j\}\). Set \[ \Xi(q)=\sum_{\substack{\Gamma\text{ has disjoint supports}\\ \bigsqcup_{\ell\in\Gamma}D_\ell=D(q)}} \prod_{\ell\in\Gamma}k_\ell(q), \qquad \|k\|_{j,A}:=\sup_x\sum_{\ell:x\in P_\ell} e^{As_\ell}\|k_\ell\|_\infty \le w_j:=e^{-p_j^{1/4}}. \tag{114}\] In particular \(\Xi=1\) if \(D(q)\) is empty. This is a mandatory cover, not an optional polymer gas. No sign or differentiability assumption is made on the individual weights.

All these lists are translation covariant and have the spin and spatial symmetries just specified. The class consists of the densities \[ e^{v|\Lambda_j|} \exp\left\{-b\mathcal E_{h,t_j} -\sum_{\text{slots}}1_X P_X-\sum_X1_Xf_X\right\}\Xi. \tag{115}\] The scalar \(v\) is the same bulk scalar on every admitted period. Assignments of anchors are made equivariantly: distribute a term equally among its permissible anchors, retaining identical complete supports on the resulting copies. Splitting an additive term this way preserves its sum; splitting a covering weight preserves its gas, since copies with the same nonempty support are mutually incompatible. Average additive anchored coefficient and error lists over the finite dihedral group. Their full sums are invariant, so this also preserves the density. If different orbit terms initially have different masks, first strengthen them to their common orbit graph and retain the exact covering correction, as in Section 5.4. Eligibility and internal compatibility tests are carried with each label, not averaged independently. All geometric rules and free kernels commute with these symmetries. Finally, after removing the total affine coefficient, orbitwise quadratic compensation subtracts multiples of \(S_o\) whose sum is zero at every anchor; it changes only the representation. These conventions justify both the individual orbit normalization and the invariant Hessian test. Masked expressions are defined to be zero off their mask; no value of an undefined extension is used there. The estimates below also apply to signed densities of this form. Positivity is used later only for the actual probability measures obtained from (107).

To prove closure, we first isolate regions with large gradients and retain a positive energy cost for them. Gaussian completion on the complement produces local fluctuation factors. We then sum connected factors, extract the seven canonical coefficients, and return the remaining terms to the error and covering classes.

Appendix 14 fixes the profiles, row ownership and complete read sets used below. It supplies an explicit core factor and proves the derivative-support and factorization assertions.

The energy identity and the massive Gaussian

Truncate the primitive kernels at radius \(r_*=\lfloor M/10^4\rfloor\) in their respective localization units, reducing this by a fixed factor if necessary for the finite compositions below. Use a subscript \(s\) for these numerical kernels. The omitted paths have exponentially small weighted sums, including their free history differences. Retain those paths as separate factors.

Use stack variables \[X=(\sqrt{c_0}\,dq,\ell_sdq),\qquad Y=\sqrt a(V-\widehat m),\qquad X'=(\sqrt{c_0}\,d'V,\ell'_sd'V),\] and put \[\zeta=(\sqrt{c_0}\,\operatorname{clip}_{t'}d'V, m'\ell'_sd'V), \qquad(\nu,w)=\mathcal M_s\zeta.\] Here clipping changes a vector of length greater than \(t'\) to the vector of length \(t'\) in the same direction. The free estimates and the masks give, entrywise, \[ |\zeta|+|w|\le Ct,\qquad |\nu|\le Ct/L. \tag{119}\] The numerical input kinetic energy plus observation energy minus the numerical output kinetic energy is the sum of the following rows: \[\begin{align*} \mathcal F_e&=S_t(r_e)+\tfrac12m_e|X_{2,e}|^2 -\nu_e\cdot X_e+\tfrac12|\nu_e|^2, &\mathcal B_Y&=\tfrac12|Y_Y-w_Y|^2,\\ \mathcal C_E&=\zeta_E\cdot X'_E-\tfrac12|\zeta_E|^2 -S_{t'}(|d'_EV|)-\tfrac12m'_E|X'_{2,E}|^2, \\ \mathcal U_E&=\zeta_E\cdot[\mathcal M_s^*(X,Y)-X']_E, &\mathcal N_i&=\tfrac12| (\mathcal N_s\zeta)_i|^2,\\ \mathcal Z_E&=\tfrac12\zeta_E\cdot [(\mathrm{Id}-\mathcal M_s^*\mathcal M_s- \mathcal N_s^*\mathcal N_s)\zeta]_E. \tag{120}\end{align*}\] This is an algebraic identity: the cross terms cancel because \((\nu,w)=\mathcal M_s\zeta\), and the displayed defect cancels the remaining quadratic terms in \(\zeta\). The free stack identities give \[ |\mathcal Z_E|\le C_Le^{-c_LM}t^2, \quad |\mathcal U_E|\le C_Lte^{-c_LM} +Ct\sum_Y e^{-c\operatorname{dist}(E,Y)}|m_Y-\widehat m_Y|. \tag{121}\]

Insert a smooth decomposition \(G_e+(1-G_e)\) at every input edge, where \(G_e=1\) for \(r_e\le t/4\) and \(G_e=0\) for \(r_e\ge t/2\). At every observation block use an analogous decomposition for \(|V_Y-\widehat m_Y|\), with thresholds \(Wt\) and \(2Wt\); \(W\) is a fixed sufficiently large constant. A complement is a primary seed. Every true output bad edge is also a mandatory primary seed. Join seeds whose radius-\(50M\) footprints meet. For each component freeze the spins within distance \(3M\) of its seeds and assign to it the numerical rows within distance \(5M\). Frozen spins retain their Haar integration. The remaining spins have the principal coordinates \[T_x=V_{[x]},\qquad \xi_x=\log(q_xT_x^{-1}),\qquad q_x=\exp(\xi_x)T_x, \quad \xi_x\in\operatorname{Im}\mathbb H.\] The retained sector tests imply \(|\xi_x|\le C_Lt<\pi/4\). Thus this logarithm is unique; an arbitrary preimage under the exponential map is not allowed.

Select all \(\mathcal F,\mathcal B\) rows farther than \(2.5M\) from every seed. On the full sector their masks are enabled, clipping is absent, their mean is on its smooth branch, and all their spins are within \(C_Lt\) of their block reference. Rotate a row to a nearby output reference and let \[D_0=(\mathsf A_s d,-\sqrt aQ),\qquad F_i^0(V)=\operatorname{Im}(R_i(0,V)V_{o(i)}^{-1}), \quad R=(X-\nu,Y-w).\] Use the affine row \(D_0\xi+F^0\). A frozen angle is smoothly cut off outside a slightly larger \(C_Lt\) chart. For an output-inventory-free call, its \(V\) extension fixes every queried output status to no flag, replaces clipping by the identity and every good output mask by one, and extends that analytic formula through the required graph gradients \(<4t'\). Cut it off smoothly only on a larger graph domain. Equality with the actual expression is recovered on the outgoing regular mask. No extension of a discontinuous output predicate through its jump is intended. Calls with output inventory instead retain their fixed output statuses and are not field-differentiated. These conventions give the bounded extensions used below on arbitrary frozen spins and output data. In exterior variables the selected affine rows are \(D\xi+F\).

A protector is a smooth cutoff in an exterior coordinate \(\xi_x\), equal to one for \(|\xi_x|\le\pi/2\) and supported strictly inside \(|\xi_x|<\pi\). It remains attached to every factor reading that protected coordinate. The attachment rules are given in Section 5.3.

The exterior quadratic \(A=D^*D\) is the principal restriction of \[d^*\mathsf A_s^*\mathsf A_s d+aQ^*Q.\] Indeed every row incident on an exterior column is selected. For a vector extended by zero into the frozen region, block Poincare gives \[\|u\|_2^2\le CL^2(\|du\|_2^2+\|Qu\|_2^2).\] Hence \(c_L\mathrm{Id}\le A\le C_L\mathrm{Id}\), uniformly in the primary geometry and the period.

For an exterior site \(x\) let \(\Omega_x\) be the exterior sites in its radius-\(r_*\) window, and set \[(\Pi_u)_{yx}=\mathbf1_{y\in\Omega_x} (A_{\Omega_x}+u)^{-1}_{yx},\qquad E_u=(A+u)\Pi_u-\mathrm{Id}.\] Exponential conjugation, followed by the resolvent identity, proves in exponentially weighted row and column sums \[ \|(A+u)^{-1}\|+\|\Pi_u\|\le\frac{C_L}{1+u},\qquad \|E_u\|\le\frac{C_Le^{-c_LM}}{1+u}. \tag{123}\] The residual lies outside its column window, so every contribution crosses its radius. The same proof for a difference of free histories retains \(\epsilon_h\). These facts also hold if the exterior is empty.

Put \[C_s=\tfrac12(\Pi_0+\Pi_0^*),\qquad \mu=-\Pi_0^*D^*F,\qquad r_s=D\mu+F.\] By (123), \(C_s\) is positive for large \(H\), and it and all its principal marginals have upper and lower bounds at fixed \(L\). The exact substitution \(\xi=\mu+gZ\) uses the single probability law \(Z\sim N(0,C_s)\) and extracts \[ (2\pi g^2)^{3n/2}\det(C_s)^{1/2}e^{-|r_s|^2/(2g^2)} \exp\left\{\tfrac12Z^*(C_s^{-1}-A)Z -g^{-1}(A\mu+D^*F)^*Z\right\}. \tag{124}\] Here \(n\) is the number of exterior sites and the determinant includes the three colors. This identity follows by expanding the square; the last exponential remains inside the expectation.

All nonlocal corrections in (124) are retained as explicit paths. With \(E_s=AC_s-\mathrm{Id}\), \[\begin{align*} C_s^{-1}-A&=\sum_{r\ge1}(-E_s)^rA,\\ \log\det C_s&=-\operatorname{Tr}\log A+ \sum_{r\ge1}\frac{(-1)^{r+1}}r\operatorname{Tr}E_s^r, \tag{125}\\ (\log A)_{xx}&=\int_0^\infty [(1+u)^{-1}\mathrm{Id}-(A+u)^{-1}]_{xx}\,du, \quad (A+u)^{-1}=\Pi_u\sum_{r\ge0}(-E_u)^r. \end{align*}\] The leading-window term is a local scalar. Extract its empty-pattern value on every site, including frozen sites, and put the local compensations into the adjacent core. Residual chains have at least one \(E\) hop; their summed envelope, with any fixed requested support exponent, is at most \[ C_L\operatorname{poly}(M,r) e^{C B r}(C_Le^{-c_LM})^r. \tag{126}\] The resolvent integral uses \((1+u)^{-2}\) for these terms.

On a stencil farther than \(6M\) from the seeds, comparison with the full empty-pattern operators gives \[ r_s=O\bigl(C_LM^D(t^2+te^{-c_LM})\bigr),\quad |d(e^\mu T)|\le Ct/L+C_LM^Dt^2,\quad |\mu|\le Ct+C_LM^Dt^2. \tag{127}\] These estimates include normalized derivatives and free-history differences. To prove them, first replace the numerical kernels and window inverse by the full empty-pattern kernels using (123). In the full calculation the first tangent of \(e^\mu T\) is exactly \(P_hV\): the stack identities make the linear residual vanish there, and \(D^*D\) is invertible. The remaining terms are quadratic on the local \(O(Mt)\) chart. The first and second fine-difference bounds for \(P_h\) give the displayed constants independent of \(L\). Everywhere, including near cores, the extended formula still gives \(|\mu|\le C_LM^Dt\).

Prepared factors and a joint integral estimate

We describe the factors before expanding products. This is necessary because estimates on separate factors would not justify their products under a common Gaussian.

The ownership rules and complete read sets are made explicit in Sections 14.1 and 14.4. A complete read support includes geometric status queries as well as numerical arguments; a ratio’s Gaussian endpoint set is only part of that support. Disjoint complete supports give the exact factorization of Lemma 37; overlapping factors use the joint estimate below.

Mark, in each primary component, the selected input edges whose true bad-bond inventory is supplied by an old covering label. Require each marked bit to be supplied once. On every other selected input primary edge retain \(\mathbf1_{r_e\le t}\). Nonprimary edges already have \(r_e\le t/2\) on the full product. Thus these local rules reproduce the old inventory constraint exactly.

For each primary component form a compulsory factor \(B_c\) from its assigned rows, with the selected affine squares subtracted. Include the negative stationary-square exponent within \(8M\) of its seeds, all primary tests and all defaults within \(14M\), the frozen normalization reciprocals, and the leading-window determinant compensations. Retain all true and contrary output-status tests needed by these formulas. In particular a component evaluated alone does not infer the absence of a nearby output seed from unspecified outside data. Lemma 10 gives \[ |B_c|\le e^{-c_Lp^2s_c+C_LM^D(1+\log g^{-1})s_c} \le e^{-c'_Lp^2s_c}, \tag{128}\] after the choices of \(P_0,H\) made below.

On every exterior site put a smooth chart profile equal to one for \(|\xi|\le\pi/4\) and zero for \(|\xi|\ge\pi/3\). It preserves the full sector integral and removes other angular preimages. In addition, attach a protector to every exterior spin read by a hard factor or by \(B_c\). These are the protectors defined before Lemma 10; each remains attached to its factor and is never opened separately. No extra hard predicate in the moving angle \(\xi=\log(qT^{-1})\) is imposed: alignment follows from the retained sector profiles, or is imposed by a smooth larger plateau. All merely measurable dependencies are through physical input spins, fixed tags, or output statuses that will not be field-differentiated.

The remaining factors are prepared as follows. Each stated exponential factor is then opened as \(1+(e^V-1)\).

1. On unassigned \(\mathcal F,\mathcal B\) rows use \(-b(|R|^2-|R^{\rm aff}|^2)/2\) with smooth angular cutoffs supported in \(|\xi|\le C_Lt\), whose plateaus contain all full-sector values. Use smooth mean branches and no clipping. The same preparation for \(-b\mathcal U\) is supported inside its smooth mean branch. Their sup and fixed normalized derivative bounds are \(C_Lg\operatorname{poly}(M,p)\).

2. The unassigned stationary squares, necessarily farther than \(8M\) from the seeds, have bound \(C_Lg^2\operatorname{poly}(M,p)\) by (127). The unassigned \(\mathcal C\) rows are zero. The \(\mathcal N,\mathcal Z\) rows have an exceptional bound \(C_Lg^{-D}e^{-c_LM}\), since \(\mathcal N X'=0\) for full kernels.

3. A canonical term or regular error is called remote if its entire projected graph is farther than \(14M\) from the seeds. For a remote term replace its hard mask by a smooth product of graph-edge profiles, one through \(t/2\) and zero from \(2t\). The resulting function is globally smooth: the error was defined through \(4t\). For a degree-\(n\) canonical term with prefactor \(b^r\) and coefficient \(a_X\), its bound is \[ C_L|a_X|g^{n-2r}\operatorname{poly}(M,p,u). \tag{129}\] For an old error it is \(C_L\operatorname{poly}(M,p,s_X)[f_X]_{k,j}\). For a nonremote term keep the actual mask and attach protectors. The same undifferentiated bounds hold; it meets a compulsory factor by the very test that made it nonremote.

4. Keep input and output kinetic truncation tails with both of their paths and every mask query. They have exceptional bounds \(C_Lg^{-D}e^{-c_LM}\) with their exponential path prices. Input tails may use the preceding smooth or protected hard prescription. On an output-empty extension use the analytic no-output-flag formula; otherwise retain the actual output status.

5. Use the determinant chains and the linear and quadratic Gaussian-ratio chains in (125) literally. A ratio exponent \(Q_\alpha\) has at most two Gaussian endpoints and \[ |Q_\alpha(Z)|\le e_\alpha \left(1+\sum_{x\in I_\alpha}|Z_x|^2\right). \tag{130}\] Even the eighth roots of \(e_\alpha\) are summable with any of the fixed support exponents required below: combine (126) with the finite-range path count in Section 15.1. The linear exponent uses \(-g^{-1}(A\mu+D^*F)\), which has the same exceptional accuracy.

6. Replace the exterior Haar Jacobian ratio by \(1+\chi(\xi)(J_{\exp}(\xi)/J_{\exp}(0)-1)\), where \(\chi\) is a smooth cutoff to \(|\xi|\le C_Lt\) and has the required plateau. Its second term has bound \(C_Lg^2\operatorname{poly}(M,p)\) because the Jacobian differs from its identity value quadratically.

7. Open the remaining edge and noise defaults, and the global exterior chart profile, as one plus their signed failure letters. For the remote noise default use \(\chi_0(|m|)G_{\mathrm{noise}}(V-m/|m|)\), where \(\chi_0=1\) above \(3/4\) and \(\chi_0=0\) below \(2/3\); evaluate the normalized mean only on this smooth branch. This equals the original default on the full sector. The failure and all its nonzero derivative supports force a large Gaussian coordinate, as proved below.

8. An old covering weight retains its own sup envelope and all its protectors. Every one of its inventory edges is a selected input primary, so the label meets a compulsory factor. Its measurable function is never differentiated.

The empty-pattern leading determinant, the Gaussian scalar powers, the Haar constant and the observation denominator have already been extracted. Their local changes were put into \(B_c\). In particular there is no unaccounted ordinary factor of order zero.

Every preparation above preserves the original full sector product: the required cutoffs are one and the smooth branches agree there. Only after these preparations do we open factors. A resulting term need not satisfy every original default, and its primary geometry is not recomputed from the values of \(Z\).

Connected sums and exact reassembly

Here is the elementary estimate used for each connected sum. Suppose primitive majorants \(a_\alpha\) on supports \(S_\alpha\) satisfy \[ \sup_x\sum_{\alpha:x\in S_\alpha} a_\alpha e^{(B+2)s_\alpha}\le\eta, \qquad |S_\alpha|\le C_LM^2s_\alpha, \qquad C_LM^2\eta\le\tfrac14. \tag{134}\] The sum of the rooted trees above a root of load \(s\) is at most \(e^s\): assuming this bound for the descendants, the unordered children have exponential sum at most \(\exp(C_LM^2\eta s)\). Induction on tree depth and monotone convergence prove the assertion. Connected sets are bounded by their spanning trees. For a hard-core Mayer logarithm the absolute connected-graph coefficient is at most the number of spanning trees. To see this, fix a total edge order and partition connected graphs according to their minimal spanning tree. Each class is an interval of optional extra edges; its alternating edge sum is zero or has absolute value one. Treat repeated labels as distinct numbered vertices before dividing by the factorial. This gives the same rooted-tree estimate.

The complete supports make the algebra match these bounds. Give each noncore read position its \(20M\) halo. A mean or covariance window reads only this halo, including the \(3M\) hole-status test. The mask radius \(O(\log(1/t))\) is smaller than \(r_*\) for large \(H\). All reads of a core, including its \(14M\) defaults and their windows, fit inside the \(50M\) footprint with these margins. Long determinant or inverse chains retain each successive window as a separate charged hop. Two disjoint complete supports therefore have zero cross covariance under \(C_s\) and disjoint frozen/tag variables. Their individual marginal entries are unchanged when outside primaries are deleted. Their integrals consequently factor exactly.

Group each connected collection of selected prepared factors into an integrated polymer, retaining its original compatibility, coverage and status checks. A disjoint family of such polymers with the full required output inventory reconstructs exactly the original selection of prepared factors. An output-empty polymer can be required to have all of its output graph good: otherwise the mandatory cover of a bad edge in that graph would intersect it and belong to the same component. It is therefore a regular masked function with the extension furnished by Lemma 11.

Apply the Mayer expansion only to these output-empty polymers and write its logarithm as \(J=\sum_i1_{X_i}j_i\). For a fixed compatible family of output-seeded polymers, the allowed background has logarithm equal to \(J\) minus just those Mayer terms whose supports meet that family. Factor out \(e^J\), open each of these forbidden terms once as an exponential-minus-one letter with the opposite sign, and join it to the seeded polymers it meets. Every new component has nonempty output inventory. We obtain exactly \[ e^{v_{\rm step}|\Lambda'|-b\mathcal E'}e^J\Xi'_{\rm pre}, \tag{135}\] where \(\Xi'_{\rm pre}\) is again a mandatory covering gas.

The same operation implements every later mask strengthening. The logarithmic correction \((1_X-1_Y)j\) for a stronger mask \(Y\) can be nonzero only if its support contains a true output bad edge. In a full family it therefore meets that edge’s covering label. Open the correction once and regroup. If supports have been enlarged, regroup by the enlarged supports while retaining the original compatibilities inside the new weights. Connected-component decomposition is a bijection, so these operations preserve the density and the inventory exactly.

For countable lists, do the algebra first for finite subsets of the prepared factors. The resulting identity is for that truncated prepared expression. Lemma 11 and (134) give a volume-exponential absolute majorant, also for the required derivatives. Dominated convergence then gives (135) for the original full expression. No positivity of the truncated signed gases is used.

The canonical coefficients

The canonical map is a finite-order algebraic operation on the full bulk kernels. It is useful to define it before comparing it with the exact integration. Use the empty primary pattern, no masks or cutoffs, and the full free kernels. Write \[R=(X-\nu,Y-w),\qquad(\nu,w)=\mathcal M X',\qquad U_E=X'_E\cdot[\mathcal M^*(0,\sqrt a(m-m/|m|))]_E.\] The full energy identity is \(\mathcal E(q)+|Y|^2/2-\mathcal E'(V)=|R|^2/2+\sum_EU_E\). Use its affine row \(R^{\rm aff}=D_f\xi+F_f\), its mean \(\mu_f=-C_hD_f^*F_f\), and its stationary residual \(R_{\mathrm{stat}}=D_f\mu_f+F_f\). Substitute \(\xi=\mu_f+gZ\), \(Z\sim N(0,C_h)\), in the following negative-log vertices: \[ \frac b2(|R_i|^2-|R_i^{\rm aff}|^2),\qquad bU_E,\qquad\frac b2|R_{\mathrm{stat},i}|^2,\qquad \text{the old canonical slots on }q,\qquad -\log\frac{J_{\exp}(\xi_x)}{J_{\exp}(0)}. \tag{136}\] An anchor is removed by common right multiplication before using its coordinates \(y_o\). Count \(y_o\) and \(gZ\) as order \(g\), and retain total order at most four in the negative logarithm of the Gaussian expectation. Equivalently, for each ordered multiset of \(v\le4\) vertices use its connected Wick expectation with coefficient \((-1)^{v+1}/v!\). Both internal and intervertex Wick pairs are included. The coefficient sums converge absolutely by the free exponential bounds and (110).

Write the resulting expression as \(\sum_{r,d}g^{-2d}F_{r,d}(y)\), with \(r\le4\). Thus \(F_{r,d}\) is the coefficient polynomial without its displayed power of \(g\), and is homogeneous of field degree \(r+2d\). Scalar polynomials are extracted per unit volume. Let \(S_{K',o}^{[n]}\) be the degree-\(n\) part, in the relative \(y_o\) chart, of the analytic unmasked output kinetic density, assigned to its anchor by splitting edge and row contributions between their endpoints. Its affine quadratic Hessian is one. Its odd-degree parts vanish, since it is quadratic in ambient spins and every dot product has even total degree in this chart.

Let \(\operatorname{Comp}\) denote the orbitwise quadratic compensation described after (110). The map is \[\begin{align*} P'_4&=F_{2,1},& \alpha&=\text{affine Hessian of }F_{2,0},& I'_2&=\operatorname{Comp}(F_{2,0}-\alpha S_{K'}^{[2]}), \\ P'_5&=F_{3,1},& I'_3&=F_{3,0},\\ P'_6&=F_{4,1},& I'_4&=F_{4,0}-\alpha P'_4-\alpha S_{K'}^{[4]},& \kappa&=\text{affine Hessian of }F_{4,-1}, \\ J'_2&=\operatorname{Comp}(F_{4,-1}-\kappa S_{K'}^{[2]}). \tag{137}\end{align*}\] Here and below the affine Hessian means its bulk sum per anchor. After the bulk affine coefficient has been removed, the sum of the individual orbit compensation coefficients is zero; thus \(\operatorname{Comp}\) does not change that polynomial sum.

Appendix 13 specifies one common coefficient norm and proves the diagonal estimate, including compensation and the translation-invariant bulk symmetry needed by the quadratic slots.

The regular error and the exact Taylor comparison

The one old-error contribution which cannot be discarded by order counting is its isolated linear response \[ \mathcal T f_X(V)=\mathbb E_{C_s} [\chi_X(q)f_X(q)],\qquad q_x=\exp(\mu_x+gZ_x)V_{[x]}, \tag{139}\] in the empty primary pattern. Give it the projected old anchor. For \(s_X\le L^{1/4}\) include a complete output good-field ball of radius \(CM'\) and its required stencils. Increasing \(C_{\mathrm{adm}}\) makes these cases nonwinding. Any strengthening of an existing mask is compensated by the exact operation following (135).

We now compare the exact logarithm in (135) with (137). An ordinary positive-order primitive of order \(r\) has majorant \(C_Lg^r\operatorname{poly}(M,p)\). Inflate it by \(g^{-.96r}\) in (134). Its remaining \(g^{.04r}\) still beats every fixed logarithmic loss and the factor \(M^2\). The connected sum of terms of total order at least five therefore retains \(g^{4.8}\). A smaller exponent pays the remaining fixed derivative and logarithmic costs. Thus, after removing the isolated old-error response, all nonprincipal terms cost \[ C_Lg^{4.6},\qquad C_Lg^{4.5}\tau\text{ for discrepancies}, \quad \tau=u+\epsilon_h+\lambda_g, \tag{143}\] in the needed anchored norms and a sufficiently large fixed support exponent. Old errors are counted as order four here but retain their stronger actual cap; every nonisolated use has total order at least five. Exceptional terms and canonical paths longer than \(M\) have arbitrary power accuracy.

For the principal terms, group the at most four origins at one anchor and strengthen their masks to a common ball. Choose \(C_\chi\) large and then \(L\) large: the old canonical balls shrink by \(L\), while the fixed number of new stencils uses only fixed multiples of \(M\). The common ball contains every required read. Its complement is compensated in the covering gas. On that ball and the even Gaussian guard, every prepared factor has its smooth vacuum formula. Taylor expansion through total order four, then restoration of full Gaussian polynomial moments, costs \(C_Lg^5\operatorname{poly}(M,p)\), also in the normalized derivative norms. The largest vertex degree required is six; a higher normalized derivative retains the same power because each direction supplies a factor \(t\).

On each fixed origin multiset the exact product and Mayer-log identities are identities of these finite polynomials. They give precisely its connected Wick expectation under the short Gaussian, including repeated origins and the Jacobian logarithm. Replacing short kernels and covariance by full ones costs exponential accuracy in \(M\), with all fixed position moments and discrepancies. Any connection that appears only after this replacement contains a long kernel path and has that same accuracy. The principal answer is therefore exactly (137), up to (143). This proves that the formal coefficient operation describes the actual integration to the claimed accuracy.

Normalization, periods, and closure

The remaining task is to remove the constant and affine-Hessian parts of the regular error without changing the density. These two extractions determine the scalar increment and the remaining kinetic increment \(\Delta\).

Put \(b^-=b+\alpha+\kappa/b\). Inserting the full canonical lists with their masks and expressing their displayed prefactors in terms of \(b^-\) leaves a raw regular list. Denote its norm by \(R\) and the norm of its difference for two inputs by \(R^\Delta\). The preceding two subsections give \[\begin{align*} R&\le q_1\delta_j+C_Lg^{4.6},\\ R^\Delta&\le q_1\delta_j^\Delta+ C_L\operatorname{poly}(M,p)\delta_j(\epsilon_h+\lambda_g) +C_Lg^{4.5}\tau, \qquad q_1=C/L+o(1). \tag{144}\end{align*}\] Here \(\delta_j^\Delta\) is the actual input error discrepancy, before division by \(\delta_j\), and \(\lambda_g=|b_2-b_1|/H_j\) up to a fixed constant. The constant multiplying \(L^{-1}\) is fixed before \(L\); the \(o(1)\) is made small by the subsequent choice of \(H\). These estimates hold with any of the finitely many larger support exponents reserved for the operations that follow.

We describe the insertion of the kinetic terms explicitly. Assign each analytic base path to an anchor, using the convention in (137), and require on its mask every edge queried by the relevant \(m'_e\). On that mask it is the analytic chord expression. Its Taylor remainder after degree four for the \(\alpha\) term, or degree two for the \(\kappa/b\) term, has the accuracy in (143). A long path pays its position and derivative powers from exponential localization. Off the mask keep the actual clipped path expression as a covering correction. Its site-hit bound is a small power of \(g\): the direct clipped energy is at most \(Ct'\), and an active second-slot row is bounded using \(m'_e=1\) and its exponentially weighted chord bound. Thus the exact insertion uses no Taylor formula outside its domain.

For a nonwinding raw error at anchor \(o\), let \(v_i=f_i(1)\) and let \(b_i\) be its unit-affine Hessian. The direction norm gives \[|v_i|\le[f_i]_{k,j-1},\qquad |b_i|\le C(t')^{-2}[f_i]_{k,j-1}.\] There is no support-size loss in the second estimate: a unit affine field at distance \(r\) is bounded by a constant times \((1+r)^8\). After strengthening the mask to include the nearest neighbors of \(o\), use the exact identity \[ 1_Xf_i=v_i+b_iS_o+ 1_X(f_i-v_i-b_iS_o) -(1-1_X)(v_i+b_iS_o). \tag{145}\] The middle term has zero value and zero affine Hessian. On its extension, \(S_o\) and its normalized derivatives are bounded by \(C(t')^2\), so its norm is at most a fixed multiple of that of \(f_i\). The last term meets an output bad bond and is included in the covering gas by the exact regrouping in Section 5.4.

Sum the bulk \(v_i\) per anchor into the volume scalar and put \(\Delta=\sum_i b_i\), also using the bulk lists. Replacing \(b^-\) by \(b'=b^-+\Delta\) leaves the negative-log correction \[ -\Delta\bigl(\mathcal E'-\sum_oS_o\bigr). \tag{146}\] Allocate the compensating \(S_o\) to each base path in proportion to that path’s affine Hessian. These Hessians sum to one. On its path mask each resulting expression has zero affine Hessian, and the sum of its regular norms is at most \(C(t')^2|\Delta|\). The constant here is independent of \(L\): it uses only the uniform exponentially weighted free-row moments and the fixed direction order. Mask complements are again covering corrections. The changes in the seven displayed canonical prefactors are smaller; for example the change \(\Delta P'_4\) has an additional factor \((t')^4\). Consequently the final regular norm and coupling error satisfy \[ \|f'\|_{k,j-1,A}\le Cq_1\delta_j+C_Lg^{4.6}, \qquad |\Delta|\le C_LH_j^{-1.05}. \tag{147}\] The same operations give the difference estimate in (144), up to fixed factors and the indicated \((t')^{-2}\) for \(\Delta\). In particular no derivative above order \(k\) of an input error has been used.

We next justify using bulk \(v_i,b_i\) on every admitted period. Every finite-period free entry is the fold of its lifted paths. All primitive choices retain complete lifted read records. A connected calculation with load below \(c_*n'/M'\) therefore agrees exactly with its bulk calculation. A disagreement has load at least \(c_*n'/M'\), and hence has an additional factor \[\exp(-c_*A_1n'/M')\] from a reserved support exponent \(A_1\). Keep a period-dependent constant or Hessian disagreement as an actual winding error, with a declared record of load \(O(n'/M')\). The extra exponential pays this record. Its constant is an allowed winding term; its Hessian is multiplied by \(S_o\), recovering the factor \((t')^2\) in the regular norm. An off-mask correction has the same covering interpretation as before. Already winding inputs remain winding because \(s'\ge4s/L\) in (109). Thus all period discrepancies stay in the error or covering lists. The extracted coupling and volume scalar are precisely the bulk ones and are independent of the period.

Here is a finite order of choices closing all the estimates. Reserve a sufficiently large fixed support exponent \(E_*\) for the finitely many tree, mask, orbit and winding operations, and let \(D_*\) bound their geometric dilations. First choose \(L\) large enough that \[E_*D_*/L<A/4,\qquad C_{\rm lead}e^{2E_*D_*}/L<1/16,\] as well as the free and core requirements already stated. \(C_{\rm lead}\) uses only the uniform free moments and the fixed derivative and reset conventions. It does not contain a constant that grows with \(L\). Choose the seven canonical caps successively using Lemma 12. Choose fixed triangular weights for their discrepancy norm.

Let \(D\) exceed every fixed polynomial degree in \(M\), support load, Gaussian score variables, and normalization entropy used above. Increase it for the finitely many remaining regroupings and choose \(P_0\ge2D+2\). Finally increase \(H\) until, uniformly for \(H_j\ge H\), \[\begin{align*} C_Le^{-c_LM}\operatorname{poly}(M)&\ll1,& C_LtM^D&\ll1,\\ C_LM^D/p^2&\ll1,& C_LM^D(T^2/t+e^{-c_LM}/t+t)&\ll1,\\ C_LM^D(1+\log g^{-1})&<c_Lp^2/4,& c_Lp^2/M^D&\gg p^{3/10}+D\log g^{-1}+D\log M,\\ C_Lg^{.04}\operatorname{poly}(M,p)&\ll M^{-2},& C_Lg^{4.6}/\delta_j&\ll1 . \tag{148}\end{align*}\] Each condition follows either from \(P_0\ge2D+2\) or from a positive power of \(H_j\) dominating every fixed power of its logarithm. For instance \(T^2/t=H_j^{-.48}/p\), and \(g^{4.6}/\delta_j=O(H_j^{-.25})\). The choices are therefore noncircular.

Equation (147) now gives the renewed error cap. Every output-seeded component retains a compulsory suppression. The reserve, the shared Gaussian tail estimate and (148) give, after all its decorations and regroupings, the stronger covering bound \(e^{-p^{3/10}}=o(w_{j-1})\). This pays the fixed hard-derivative losses and the relative input error and gas discrepancy norms. Symmetrizing the assignments by the finite spatial group preserves these bounds and the density.

For completeness, divide the error discrepancy in (144) by \(\delta_{j-1}\). The remainder factor is small because \(g^{4.5}/\delta_j=O(g^{.4})\). The gas has just been bounded with strictly more suppression than its required relative cap. The canonical map is triangular with contracting diagonal. Choose the later discrepancy weights sufficiently small, and then enlarge \(H\) for the remaining cross terms. This gives a common \(q<1\). Dependence on the precise coupling costs \(\operatorname{poly}(M,p)|b_2-b_1|/H_j\), which is at most \(C_LH_j^{-1/2}|b_2-b_1|\) after increasing \(H\). Differences of the canonical kicks cost \(C_Lu\); differentiating \(\kappa/b\) costs \(C_L|b_2-b_1|/H_j^2\); the extracted raw Hessian has the bounds just proved. Weakening the positive decay exponent if necessary gives (117)–(118). This finishes the proof of Theorem [thm:rg-closure].

A finite-torus expansion

For this calculation only, take spins on \(S^D\), so that there are \(D\) tangent components. Let \(v=n^2\). Write \(A(n)\) and \(Z_2(n)\) for the coefficients of \(\beta^{-1}\) and \(\beta^{-2}\) in \(\log Z_\beta(n,n)\) at fixed \(n\). A subscript \({\rm d}\) denotes the operation \(F_{\rm d}(n)=4F(n)-F(2n)\), and put \(D_0=3D/2\).

Near an aligned configuration choose the mean-spin axis and write \[q_x=(\sqrt{1-|u_x|^2},u_x),\qquad \sum_xu_x=0, \qquad z_x=\sqrt\beta\,u_x.\] Integrating over the mean axis gives the chart density \[\prod_x(1-|u_x|^2)^{-1/2} \left(v^{-1}\sum_x\sqrt{1-|u_x|^2}\right)^D.\] Indeed the delta constraint that the mean tangent component vanishes has axis Jacobian \(|\bar q|^{-D}\), so the displayed factor cancels that Jacobian. The root with positive mean component is selected. At fixed volume every configuration outside a neighborhood of the aligned manifold has a positive action excess, and hence contributes exponentially little as \(\beta\to\infty\).

Let \(K\) be the nearest-neighbor Laplacian and let \(G_x\) be its mean-zero inverse on the torus. In this subsection put \[G=G_0,\quad M=v^{-1}\sum_xG_x^2,\quad r=\frac{1-v^{-1}}2,\quad c=G-r/2=G_{e_1}.\] The components of \(z\) are independent centered Gaussian fields with covariance \(G\). Put \(R_x=|z_x|^2\). Expansion of the action and chart density gives \[\begin{align*} L_1={}&\frac12(1-D/v)\sum_xR_x -\frac18\sum_{\langle xy\rangle}(R_x-R_y)^2, \tag{151}\\ L_2={}&\left(\frac14-\frac D{8v}\right)\sum_xR_x^2 -\frac D{8v^2}\left(\sum_xR_x\right)^2 -\frac1{16}\sum_{\langle xy\rangle} (R_x-R_y)^2(R_x+R_y). \tag{152}\end{align*}\] All edge sums in these two formulas are unoriented. Therefore \[A(n)=\mathbb EL_1,\qquad Z_2(n)=\mathbb EL_2+\tfrac12\operatorname{Var}L_1.\]

The first coefficient is \[ A(n)=\frac{Dv r^2}{4}-\frac{D(D-1)}2G. \tag{153}\] For example \(\mathbb E(R_x-R_y)^2=4D(G^2-c^2)\) on an edge, which gives (153) directly. For the next coefficient, the independent Gaussian variables \((z_x+z_y)/\sqrt2\) and \((z_x-z_y)/\sqrt2\) have covariances \((G+c)I\) and \((G-c)I\). Their moments give \[\begin{align*} \mathbb EL_2={}&D(D+2) \left[\left(\frac v4-\frac D8\right)G^2-vrG^2+\frac{vr^2G}{4}\right] -\frac{D^3G^2}{8}-\frac{D^2M}{4}. \tag{154}\end{align*}\]

To compute the variance, decompose \(L_1\) into its second and fourth Gaussian chaoses, that is, its normal-ordered homogeneous Gaussian polynomials. The quadratic coefficient matrix of its second chaos is \[-\tfrac12[cK+(D-1)I/v].\] This follows by replacing the second-chaos part of \((R_x-R_y)^2\) by \(4G(:R_x:+:R_y:)-8c:z_x\cdot z_y:\) and summing the four incident edges. Since \(KG\) is the projection off constants, its contribution to half the variance is \[ \frac D4\{c^2(v-1)+2c(D-1)G+(D-1)^2M\}. \tag{155}\] The fourth chaos is \(-\frac18\sum_{xy}K_{xy}:R_xR_y:\). In its covariance there are four pairings which send all fields at \(x\) to one endpoint and all fields at \(y\) to the other, four with the endpoints exchanged, and sixteen mixed pairings. The first eight have color factor \(D^2\) and the last sixteen factor \(D\). Thus its contribution to half the variance is \[ v\left(\frac{D^2T_1}{16}+\frac{DT_2}{8}\right), \tag{156}\] where \[\begin{align*} T_1&=\sum_x\bigl(KG_\bullet^2\bigr)_x^2,\\ T_2&=\sum_yK_{0y}\sum_xG_xG_{x-y} \bigl(K(G_\bullet G_{\bullet-y})\bigr)_x. \end{align*}\] Distinct chaoses are orthogonal, so no further variance terms occur.

Here are useful reductions of these expressions. In sums indexed by \(a,d\), let these indices run over all four nearest-neighbor directions. Set \[J_a(x)=G_{x+a}-G_x,\quad P(x)=\sum_aJ_a(x)^2, \quad h_{ad}(x)=J_d(x+a)-J_d(x),\] and \[j_0=\sum_xP(x)^2,\qquad j_1=\sum_{x,a,d}J_a(x)^2h_{ad}(x)^2.\] The discrete product rule \(K(fg)=fKg+gKf-\sum_a(\nabla_af)(\nabla_ag)\) gives \[\begin{align*} \sum_xG_xP(x)&=G^2-M,\tag{157}\\ T_1&=4G^2(1-v^{-1})-4Gr^2+j_0,\tag{158}\\ T_2&=8rG^2-2r^2G-2(G^2-M)/v-W,\tag{159}\\ W&=2r(G^2+c^2)+j_0/2-j_1/4. \tag{160}\end{align*}\] For clarity, (160) follows by applying the product rule to \(W=\sum_{x,d}G_xJ_d(x)(K(G_\bullet J_d))_x\). Its three terms are \[\sum_xG_x(KG)_xP(x),\quad \sum_{x,d}G_x^2J_d(x)(KJ_d)_x, \quad-\sum_{x,a,d}G_xJ_a(x)J_d(x)h_{ad}(x).\] The middle term is \(2r(G^2+c^2)\). In the last term use \(2J_dh_{ad}=J_d(x+a)^2-J_d(x)^2-h_{ad}^2\) and sum by parts. The difference cancels the first term and supplies \(j_0/2\); reversal of each \(a\)-edge gives \(\sum G_xJ_a h_{ad}^2=-j_1/2\), supplying \(-j_1/4\). The other three identities follow from the same product rule and \(KG=\delta_0-v^{-1}\).

Combining (154)–(160) yields \[ Z_2(n)=Dv\left[\frac{D-1}{16}j_0+\frac1{32}j_1+\frac1{128}\right] -\frac{D(D-1)^2}{4}G^2-\frac{D(D-1)}8G+O(1). \tag{161}\] In fact the remainder displayed as \(O(1)\) is exactly \[\frac{D(D-1)(D-2)}4M+\frac{D(D-1)}{8v}G -\frac{3D}{128}+\frac{3D}{128v}-\frac D{128v^2}.\] This also verifies that the remainder is bounded, using \(G=O(\log n)\) and \(M=O(1)\).

The logarithm in the finite-size correction

Let \[G^\infty(x)=\int_{[-\pi,\pi]^2} \frac{\cos(p\cdot x)-1}{\sum_\mu4\sin^2(p_\mu/2)} \frac{\mathrm d^2p}{(2\pi)^2}\] be the infinite-lattice potential, with \(G^\infty(0)=0\), and use superscript \(\infty\) for its differences. For \(d=1,2,3\) one has, with \(x\) in the centered fundamental square for the torus term, \[\begin{align*} |\nabla^dG(x)|+|\nabla^dG^\infty(x)|&\le C(1+|x|)^{-d}, \tag{162}\\ |\nabla^d(G-G^\infty)(x)|&\le Cn^{-d}\quad(|x|_\infty\le n/3). \tag{163}\end{align*}\] The bound for \(G^\infty\) holds at every lattice point. To see these bounds, partition the Fourier multiplier \([\sum_\mu4\sin^2(p_\mu/2)]^{-1}\) into smooth dyadic annuli of radius \(s\). After \(d\) differences, integration by parts bounds each annulus by \(C_Ns^d(1+s|x|)^{-N}\). The corresponding torus sum is its periodization. For \(s\gtrsim n^{-1}\) sum its nonzero translates; for smaller \(s\) sum the infinite-lattice estimates directly. These give (162)–(163). The same Fourier comparison gives \(G=(2\pi)^{-1}\log n+O(1)\) and \(M=O(1)\). The leading symbol \(ia\cdot p/|p|^2\) gives \[J_a^\infty(x)=-\frac{a\cdot x}{2\pi|x|^2}+O(|x|^{-2}).\] One may evaluate it by replacing a cutoff at zero by \(e^{-|p|^2}\); the difference has the stated faster decay, and \(e^{-|p|^2}/|p|^2=\int_1^\infty e^{-t|p|^2}\,\mathrm dt\).

The equation and square symmetry improve the local comparison. At zero its first difference is \(1/(4v)\), its pure centered second difference is \(1/(2v)\), and its mixed centered second difference is zero. Summing the third-difference bound therefore gives \[ J_a(x)-J_a^\infty(x)=\frac{a\cdot x}{2v}+\frac1{4v} +O((1+|x|)^2/n^3),\qquad |x|_\infty\le n/4. \tag{164}\] Put \(c_*=(2\pi)^{-1}\). For \(1\le|x|\le n/4\), the preceding formulas imply \[\begin{align*} P(x)-P^\infty(x) &=-2c_*/v+O((v|x|)^{-1}+|x|/n^3+|x|^2/v^2),\\ P^\infty(x)&=2c_*^2/|x|^2+O(|x|^{-3}). \end{align*}\] In the difference of the squares the only logarithmic sum is \(-8c_*^3v^{-1}\sum_{1\le|x|\le n/4}|x|^{-2}\). All the displayed error terms, and the tails outside this range, sum to \(O(v^{-1})\). For \(j_1\) the infinite summand decays as \(|x|^{-6}\) and its differentiated comparison errors are summable. Consequently \[ j_0=j_0^\infty-\frac{2\log n}{\pi^2v}+O(v^{-1}),\qquad j_1=j_1^\infty+O(v^{-1}). \tag{165}\]

Volume terms cancel in the doubling operation. Substituting (165) into (153), (161) gives \[\begin{align*} A_{\rm d}(n)&=-\frac{3D(D-1)}{4\pi}\log n+O(1), \tag{166}\\ Z_{2,\rm d}(n)+\frac{A_{\rm d}(n)^2}{2D_0} &=-D_0\frac{D-1}{4\pi^2}\log n+O(1). \tag{167}\end{align*}\] There is an instructive cancellation in the second formula. Write \(G'=G_{2n}(0)\). The quadratic Green terms reduce to \(D(D-1)^2(G-G')^2/3\), which is bounded. The cross term in \(A_{\rm d}^2/(2D_0)\) cancels the linear Green term in \(Z_{2,\rm d}\). Thus the only unbounded term is the finite-size correction to \(j_0\).

Computing the same coefficients after blocking

We now set \(D=3\). Choose a fixed sufficiently large dyadic \(m\) and put \(n=mL^s\). Insert \(s\) normalized observation kernels at parameters \(\beta_h=\beta+B_h+C_h/\beta\), on both the \(n\) and \(2n\) tori. Pin one spin on the last layer to remove global rotation. For each fixed \(s\) this is a finite-dimensional integral with the same partition function up to a coupling-independent constant. Here \(\beta\) tends to infinity at fixed \(s\); the calculation does not require an admitted trajectory in the reference bands \([H_j/2,2H_j]\). Its aligned saddle is unique after pinning, with strictly positive transverse quadratic form. All other configurations have a positive action excess: zero microscopic energy forces alignment, and zero observation energy propagates that alignment through every layer, including the fallback branch of the observation.

Successive Gaussian completion and Wick integration compute exactly the same finite Taylor coefficients as direct Laplace expansion. These are the canonical prescriptions of Lemma 12. Replacing the next formal parameter by \(\beta+B_{h+1}+C_{h+1}/\beta\) changes the kinetic coefficient only by \(O_s(\beta^{-2})\); its two angular factors put that change beyond the field order in question. Scalar powers are still retained at their specified order.

We justify the uniform error needed in comparing coefficients as \(s\) varies. At a step whose output period is \(\bar m=mL^{s-h-1}\), a finite-period coefficient differs from its folded bulk value only if the complete contraction paths travel a distance at least \(c\bar m\). Analytic coefficient norms therefore bound this discrepancy by \(C_Le^{-c\bar m}\) per anchor. A previously introduced discrepancy of size \(\delta\) in the ordinary coefficients grows by at most \(C_L\) in one formal step: there are at most four fluctuation orders, and all the coefficient and covariance sums are absolutely summable. There is no need to impose an affine cancellation on this discrepancy. Earlier periods are larger by successive factors \(L\), so \[\sum_{r\ge0}C_L^{r+1}e^{-cmL^r}<\infty.\] The same sum with any fixed polynomial in \(r\) and \(mL^r\) is finite. The period factor pays for the sum over anchors, including scalar Wick contractions. The depth factor also pays for reexpressing their inverse-coupling powers in terms of \(\beta_s\), since \(|B_s-B_h|\le C_L(s-h)\). Thus both inverse-coupling coefficients of the propagated winding contributions are bounded uniformly in \(s\).

At the final fixed period \(m\), the pinned Gaussian and canonical coefficients are bounded uniformly in \(s\). All bulk scalar powers at earlier stages are proportional to volume and cancel in the doubling operation. The remaining Gaussian pinning power is \(D_0\log\beta_s\): the powers on periods \(m\) and \(2m\) are respectively \(-D(m^2-1)/2\) and \(-D(4m^2-1)/2\), whose doubling difference is \(D_0\). The blocked expansion consequently has the form \[\begin{gathered} \Delta_\beta(mL^s) =D_0\log\beta_s+\mathcal C_s +\frac{X_s}{\beta_s}+\frac{Y_s}{\beta_s^2} +O_s(\beta_s^{-3}),\\ |X_s|+|Y_s|\le C_{L,m}. \end{gathered}\] Here \(\mathcal C_s\) is independent of \(\beta\), and \(X_s,Y_s\) include the final fixed-period coefficients and all propagated winding terms. Their bounds are uniform in \(s\); the displayed remainder is a fixed-\(s\) Laplace remainder. Substituting \(\beta_s=\beta+B_s+C_s/\beta\) and comparing coefficients gives \[\begin{align*} A_{\rm d}(mL^s)&=D_0B_s+X_s,\tag{168}\\ Z_{2,\rm d}(mL^s)&=D_0(C_s-B_s^2/2)-B_sX_s+Y_s. \tag{169}\end{align*}\] Equations (166), (168) prove the first part of (149). Adding \(A_{\rm d}^2/(2D_0)\) in (169) cancels both \(B_s^2\) and \(B_sX_s\); then (167) proves its second part.

The triangular contraction of the canonical coefficients, with the forcing \(O(L^{-h})\) from the free kernels, gives exponential convergence of each kick. The bounded cumulative formulas (149) identify their limits, proving (150) and the theorem. Notice that no uniform bare Laplace remainder was used.

Length as a function of the bare coupling

For later use we record the consequence for the exact trajectories.

Initialization and admitted trajectories

Here \(j\) counts remaining layers: \(b_N\) is the bare coupling and \(b_0\) is terminal. The chronological free-history index is \(h=N-j\), as in the coefficient calibration and the introductory drift formula.

At free history zero, the unmasked quadratic energy is the nearest-neighbor energy. Its representation in the class has all seven slots and all regular errors zero. To construct the covering gas, group the true bad bonds into connected components after padding by a fixed multiple of \(R_*\log(100/t)\). Assign every deleted mask row to a bad bond witnessing its deletion, and retain that witness in the component record. The difference between the original energy and the clipped energy is nonnegative. Each bad bond supplies at least \(cb r_e^2\ge cp^2\) to this difference. The remaining deleted rows are nonnegative squares. The weight of a component is the exponential of minus its assigned difference, with the indicators recording its exact bad-bond inventory. These components have disjoint padded supports, or are joined when their supports meet. Their product therefore gives exactly the original density.

Connected positioning and padding cost at most \(\exp(Cs\log M)\) for a record of load \(s\). The retained \(p^2\) reserve dominates this count and \(e^{As}\) by (148), proving the initial covering cap. The initial weights are continuous in the precise bare coupling: their geometric predicates use the fixed reference parameters \(H_j,t_j,\mathfrak m_j\), so only the exponential energy factors vary. The retained reserve makes this continuity uniform in the covering norm on each admitted coupling interval. For two nearby inputs with the same free history, the comparison estimates (117)–(118) have \(\epsilon_h=0\) and preserve continuity of both the shape data and the extracted coupling. Induction therefore gives a continuous terminal-coupling map on every finite sequence of strictly admitted inputs. This is the continuity used in the shooting argument below.

Write \(\alpha_h,\kappa_h\) for the kicks on the canonical orbit started from zero. Theorem 14, proved in Section 6, gives \[ \alpha_h=-\gamma+O_L(\rho_c^h),\qquad \kappa_h=-\frac{\gamma}{2\pi}+O_L(\rho_c^h), \qquad 0<\rho_c<1 . \tag{171}\] The regular error \(\Delta\) is not used in that formal calculation. It is bounded for the actual trajectory by Theorem [thm:rg-closure].

Matching at a common terminal coupling

Figure 2 displays the two boundary conditions used in the following comparison.

The endpoint densities in this theorem are strictly positive: they are repeated integrals of the positive microscopic density against the strictly positive kernel (107). More quantitatively, changing \(B\) endpoint spins changes the logarithm of the true endpoint density by at most \(C_LH|B|\). For each fixed input the last observation densities have this ratio bound, since their coupling is in a fixed band of order \(H\) and chord lengths on \(S^3\) are bounded. Integrating their ratio inequality against the positive preceding-layer measure preserves it. This positivity statement concerns the complete density; the individual covering weights remain allowed to have either sign.

The finite-volume comparison problem

Let \[\Lambda_M=(\mathbb Z/M\mathbb Z)^2,\qquad v=|\Lambda_M|=M^2, \qquad q\in(S^3)^{\Lambda_M},\] where \(M\) is dyadic. Write \(\,\mathrm dq\) for product normalized Haar measure, \(r_e=|q_x-q_y|\) for \(e=\langle xy\rangle\), and \[D(q)=\{e:r_e>t_H\}.\] The large parameter \(H\) is the inverse coupling at the retained scale. It is fixed throughout a comparison. In the application, \[t_H=H^{-1/2}p_H,\qquad p_H=(\log H)^{P_0},\] with the fixed exponent \(P_0\) chosen in the renormalization construction.

The word mandatory refers to the equality of inventories in (177). In particular, \(\Xi\) is not the partition function of an optional dilute gas.

Consider two pointwise positive densities \[ \rho_i(q)=e^{X_i(q)}\Xi_i(q)>0,\qquad Z_i=\int\rho_i(q)\,\,\mathrm dq,\qquad \,\mathrm d\mu_i=Z_i^{-1}\rho_i\,\,\mathrm dq,\quad i=1,2 . \tag{178}\] Use a common countable label set, putting unused weights equal to zero. The functions in (178) are the pointwise versions provided by the exact blocking identity.

Here are the precise data needed for comparison. A quantity described as polynomial in \(H\) is bounded by \(CH^a\), with \(C,a\) fixed independently of the period and the cutoff depths.

1. Each \(P_\lambda\) is contained in a connected lattice graph with at most \(R_Hs_\lambda\) vertices, where \(R_H\) is polynomial in \(H\). For a fixed \(A\ge16\), \[\begin{align*} \sup_x\sum_{\lambda:x\in P_\lambda} e^{As_\lambda}\|k_{i,\lambda}\|_\infty &\le w_H, \tag{179}\\ \sup_x\sum_{\lambda:x\in P_\lambda} e^{As_\lambda} \|k_{2,\lambda}-k_{1,\lambda}\|_\infty &\le D_H\delta_K. \tag{180}\end{align*}\] Here \(w_H\) decreases faster than every inverse power of \(H\); \(D_H<\infty\) is independent of the period and the cutoff depths.

2. If \(q,\widetilde q\) differ at \(n\) sites, then \[\begin{align*} |\log\rho_i(q)-\log\rho_i(\widetilde q)| &\le B_{\rho,H}n, \tag{181}\\ |X_i(q)-X_i(\widetilde q)| &\le B_{X,H}n, \tag{182}\end{align*}\] where \(B_{\rho,H},B_{X,H}\) are polynomial in \(H\). Moreover \[ \|X_2-X_1\|_\infty\le D_Hv\delta_K. \tag{183}\]

3. For some \(c_*>0\) independent of \(H\), \[ X_i(q)\le \epsilon_Hv- c_*H\sum_e\min(r_e^2,t_H^2), \qquad \epsilon_H=o(Ht_H^2). \tag{184}\] On the cap \[{\cal C}_H=\{q:|q_x-\mathbf1|\le cH^{-1/2} \text{ for every }x\},\] one has \(D(q)=\varnothing\) and \[ X_i(q)\ge-\epsilon'_Hv,\qquad \epsilon'_H=o(Ht_H^2). \tag{185}\]

4. Both positive measures are translation invariant and reflection positive at the axis seams used to reflect lattice tiles. The reflection prescriptions are compatible with the tile events below. Finally, \[t_H\longrightarrow0,\qquad t_H^{-1}\le\operatorname{poly}(H),\qquad \frac{Ht_H^2}{(\log H)^2}\longrightarrow\infty . \tag{186}\]

All comparisons below are uniform over families satisfying these data. The last subsection verifies the data for the retained densities of Theorem [thm:rg-closure].

A joint rough-bond estimate under each positive law

Choose a fixed constant \(C_1\), to be enlarged in the geometric construction, and define the wider set of flagged bonds \[ \widehat D(q)=\{e:r_e>t_H/C_1\}. \tag{191}\] The distinction between \(D\) and \(\widehat D\) leaves room to modify a configuration near its true rough bonds.

Removing a rough cluster

We first record the target-space fact used in the modification.

Fix the regions and the modified values using the original configuration, once and for all. Write \({\cal I}\) for the retained regions and \(q(I)\), \(I\subset{\cal I}\), for the configuration in which precisely the regions outside \(I\) have been modified. Then \[ D(q(I))=\bigsqcup_{C\in I}D_C. \tag{197}\] The modification need not remove the wider flags at threshold \(t_H/C_1\). We never recluster a modified configuration; its geometry, its \(n_C\)’s, and its inventories are those fixed from the original one.

Put \(\Xi_i(I)=\Xi_i(q(I))\). Since \(\rho_i>0\), these are complete positive sums. Equations (181) and (182) give the crucial denominator estimate: \[ 0<\frac{\Xi_i(I\setminus B)}{\Xi_i(I)} \le \exp\!\left(C_{3,H}\sum_{C\in B}n_C\right), \qquad C_{3,H}=(B_{\rho,H}+B_{X,H})R_{E,H}. \tag{198}\] Indeed \(\log\Xi_i=\log\rho_i-X_i\), and at most \(R_{E,H}\sum_{C\in B}n_C\) sites have changed.

Exact identities for the signed covering sums

At a fixed active set \(I\), take a full original covering family and join two of its labels whenever their supports meet the same active region \(E_C\), \(C\in I\). Call a connected group a macrolabel \(U\), and give it support \[ S_U=\bigcup_{\lambda\in U}P_\lambda \ \cup\ \bigcup_{\substack{C\in I:\\ E_C\cap P_\lambda\ne\varnothing \text{ for some }\lambda\in U}}E_C. \tag{199}\] Its weight is \[w_i(U;q(I))=\prod_{\lambda\in U}k_{i,\lambda}(q(I)). \tag{200}\] Every active region it meets is covered in full by this macrolabel: all labels supplying its inventory meet that region and so belong to the same group. Different macrolabels have disjoint enlarged supports. Conversely, compatible macrolabels covering all active inventories recover exactly the original full covering families.

For a site set \(S\) disjoint from the active editable regions, let \(\Xi_i(I;S)\) be the full-inventory sum restricted to macrolabels whose supports avoid \(S\). This restricted sum may have either sign. If \({\cal A}\) is a compatible selected macrolabel family, denote its combined support by \(S_{\cal A}\), and the active regions it covers by \(B_{\cal A}\).

Introduce the nonnegative envelopes \[p_\lambda=\|k_{1,\lambda}\|_\infty+ \|k_{2,\lambda}\|_\infty,\qquad d_\lambda=\|k_{2,\lambda}-k_{1,\lambda}\|_\infty\] and the restricted ratio \[R_i(I,S)=\frac{|\Xi_i(I;S)|}{\Xi_i(I)}.\] Equations (198) and (202) give \[ R_i(I,S)\le1+ \sum_{\substack{{\cal A}\ne\varnothing,\ {\rm compatible}\\ S_U\cap S\ne\varnothing\ \forall U\in{\cal A}}} \left[\prod_{\lambda\in\bigcup{\cal A}}p_\lambda\right] e^{C_{3,H}\sum_{C\in B_{\cal A}}n_C} R_i(I\setminus B_{\cal A},S_{\cal A}). \tag{204}\] A telescoping of each finite product also gives \[ |w_2(U)-w_1(U)| \le\sum_{\lambda\in U}d_\lambda \prod_{\substack{\kappa\in U\\\kappa\ne\lambda}} p_\kappa . \tag{205}\] Thus (203) starts the same recursion with a nonempty first generation and one marked, differenced original label in each of its macrolabels. Every generation consumes at least one active region. The recursion is consequently finite.

The common exceptional set

Let \(C_{4,H}>0\) be a polynomial in \(H\), to be fixed in the next subsection. Define \(G\) by the following requirement. Take any finite collection of distinct original label nodes and distinct editable-region nodes whose physical supports form a connected intersection graph. Write \[s=\sum_{\lambda}s_\lambda,\qquad n=\sum_C n_C.\] Whenever \(n>0\), require \[ (C_{3,H}+C_{4,H})n\le \frac A4s+\zeta_HK. \tag{206}\] This quantification uses the whole common label set, not just labels eligible at the original configuration. The recursion can evaluate a label at a different \(q(I)\), and the same geometric inequality must remain available there. All these collections are countable, so \(G\) is measurable.

A tree bound for the signed recursion

We now prove the normalized estimate, including its combinatorial multiplicity.

From relative densities to partition functions

Verification for the renormalized \(O(4)\) densities

We now check the assumptions used in the theorem, including those that come from positivity of the original model rather than from bounds on its signed expansion.

At the retained scale, Theorem [thm:rg-closure] gives \[ \rho_i=e^{X_i}\Xi_i,\qquad X_i=-h{\cal E}_i-{\cal P}_i-{\cal L}_i,\qquad H/2\le h\le2H. \tag{216}\] The two densities are written using the prescribed bulk scalar normalization. The kinetic form is \[ {\cal E}_i(q)=\sum_e S_t(r_e) +\frac12\sum_e m_e(q)|(\ell_i\nabla q)_e|^2, \qquad S_t(r)= \begin{cases} c_0r^2/2,&r\le t,\\ d_0tr,&r>t, \end{cases} \tag{217}\] where \(0<d_0<c_0/16\). In this formula \(\nabla q\) denotes the field of nearest-neighbor spin differences. The free kernels \(\ell_i\) have the stated uniform exponentially weighted row and column bounds. Every mask and every other condition used by a term is part of its complete support.

Alignment and the microscopic observables

The finite-torus conclusion passes to the periodic infinite-volume state at each fixed depth \(N\). To justify this even for a measurable fallback rule, approximate the bounded local functions \(F_X\) and their products by continuous functions in product-Haar \(L^1\). The nearest-neighbor model has finite-support marginal densities bounded uniformly in the surrounding torus, with a bound depending only on that fixed support and its fixed bare coupling; this follows by bounding the finitely many incident interaction terms in its conditional density. The approximation is therefore uniform in the torus size. Local convergence then proves the claim. This passage uses no continuum construction.

Transfer across the actual support gap

The choice of \(H\) at the start of Section 9 already ensures \(\varepsilon_H\le1/9\). It remains to compare the correlation (227) with the distance between the two microscopic supports.

We will use the following direct consequence of the full transfer gap. If real centered bounded observables \(F,G\) are supported in time slabs \([-r,0]\) and \([d,d+s]\), respectively, then \[ |\operatorname{Cov}(F,G)|\le\left\lVert F\right\rVert_\infty\left\lVert G\right\rVert_\infty e^{-m_{\mathrm{lat}}d}. \tag{228}\] Indeed let \(\theta\) reflect in the zero-time plane, and translate \(G\) by \(-d\). Their half-space vectors satisfy \(\operatorname{Cov}(F,G)=\langle[\theta F],T^d[\tau_{-d}G]\rangle\). Each vector has norm at most its sup norm, by the reflected expectation defining that norm. Both vectors are perpendicular to the vacuum. Applying the definition of \(m_{\mathrm{lat}}\) proves (228). Bounded measurable observables follow by local \(L^2\) approximation and Cauchy–Schwarz. This argument uses the actual infinite-volume Hilbert space and makes no claim that finite-cylinder eigenvalues converge individually.

What would pass to an existing continuum theory

We state separately the precise spectral implication if an appropriate continuum limit is available. This is not used to prove (230).

The norm convergence can instead be imposed after positive-time regularization. More precisely, for each \(u\) in a dense subset of \(\Omega^\perp\), suppose there are centered lattice vectors \(v_j\) and some \(s>0\). Define their autocorrelations by \[C_j(t)=\langle v_j,T_j^{\lfloor t/a_j\rfloor}v_j\rangle \qquad(t\ge0),\] and assume that \[C_j(s+r)\longrightarrow\langle u,e^{-r\mathsf H}u\rangle \quad(r\ge0).\] The case \(r=0\) gives \(C_j(s)\to\|u\|^2\). For \(t>s\), positivity of the lattice spectral measure gives \(C_j(t)\le e^{-(m_j/a_j)(t-s+o(1))}C_j(s)\). Apply this with \(t=s+r\) and pass to the limit to obtain the same quadratic-form bound as in the proposition. Thus no bound on \(\|v_j\|\), or on an unnormalized field renormalization constant, is needed. Euclidean covariance is a separate condition if the energy gap is to be interpreted as a relativistic mass gap.

An upper bound needs a surviving observable. Low-energy cutoff states can disappear from a chosen limiting Hilbert space: the positive energy measures \(j^{-1}\delta_1+(1-j^{-1})\delta_R\), \(R>1\), have spectral bottom one but converge to \(\delta_R\). A sufficient observable-survival condition is the following: in the same limiting theory there are centered vectors \(f,g\), with \(\|f\|,\|g\|\le1\), and a time separation \(d\ge\ell_{\mathrm{ref}}\), such that the separated cutoff block cross correlations converge to \(\langle f,e^{-d\mathsf H}g\rangle\ge1/8\). Then, writing \(m_{\mathrm{cont}}\) for the spectral bottom on \(\Omega^\perp\), \[1/8\le |\langle f,e^{-d\mathsf H}g\rangle| \le e^{-dm_{\mathrm{cont}}} \le e^{-\ell_{\mathrm{ref}}m_{\mathrm{cont}}},\] so \(m_{\mathrm{cont}}\le(\log8)/\ell_{\mathrm{ref}}\). The mixed-correlation convergence and norm bounds are additional hypotheses; separate convergence of observables without these properties would not suffice. These convergence requirements identify the additional input for a statement about a continuum spectrum. The finite and positive mass bounds in physical units have already been proved without that input.

Domains, sampling, and the free precision

We use integer site coordinates for Fourier transforms. Changing to block-centered coordinates conjugates the formulas by deterministic translation phases; in coarse momentum these phases and their fixed derivatives are uniformly bounded. The integer-coordinate convention makes the gluing of aliases explicit. For \(\delta>0\), put \[\Omega_\delta=\{p\in\mathbb C^2: |\mathop{\rm Re}p_\mu|<\pi+\delta,\quad |\mathop{\rm Im}p_\mu|<\delta\}.\] An alias field is a family on these overlapping cubes, with the alias index relabeled when \(p\) crosses a face of the real momentum cube. Equivalently it is a function of \(k\) on the fine momentum torus, with \(p=Lk\) reduced modulo \(2\pi\). At a centered representative \(l\) write \(\langle l\rangle=1+|l|_1\). The weights of two representatives used on an overlap are comparable by an absolute constant. All estimates on overlapping cubes therefore define estimates of the same field. An analytic star means \(f^*(z)=\overline{f(\bar z)}\) for real-kernel matrix symbols, with transposition included for matrices; for the scalar Fourier transform of real weights it is \(b^*(z)=b(-z)\).

The two divisions in the edge intertwiner

The following elementary analytic fact specifies the divisions needed in the construction of \(T\). It is useful to keep the alias scale \(L^{-1}\) in its statement.

We verify the hypotheses for the actual construction. On a nonprincipal alias, Lemma 32 and its explicit formula imply \(|K_h(k_l)^{-1}|\le CL^2\langle l\rangle^{-2}\) on the coarse complex cube. The bump estimate (231) and (96) give arbitrary fixed alias decay for \(P_l\). On the principal alias the exact identity \[P_0=b(k_0)^{-1}\left[1- \left(a^{-1}+L^{-2}\sum_{l\ne0}b(k_l)b(k_l)^*K_h(k_l)^{-1} \right)K_{h+1}(p)\right]\] removes the pole. The principal bump is nonzero on the fixed smaller domain by the preceding principal-factor estimate. Consequently \(d_lP_l\) has bound \(C_sL^{-1}\langle l\rangle^{-s}\). The finite geometric sum defining \(\omega_{\mu,l}\) is bounded by \(CL\) on that domain; hence \(T_{0,l}\) has the same bound. The identities \(KP=Q^*K'\) and \(\omega_\mu d_\mu=d'_\mu\) give \(d_l^*R_l=0\) exactly. At \(p=0\) both terms in \(R_l\) vanish on nonprincipal aliases. On the principal alias their linear terms are both \(ip/L\), so \(R_0\) vanishes through first order. Lemma 33 therefore gives the stated \(\gamma,H\). In \[T_l=T_{0,l}+c_l^*H_l(B')^{-1}\] the extra factor \(c_l^*\) has size \(C\langle l\rangle/L\); using one further decay order proves \(|T_l|\le C_sL^{-1}\langle l\rangle^{-s}\). The construction is linear in \(R\) and uses uniformly bounded analytic inverses, so its history differences have the same \(L^{-h}\) factor. Choose \(J\) once after listing all required decay and difference orders.

Finally, the remaining factorizations in the positive completion introduce no further scale loss. A coarse analytic row \(v\) with \(vd'=0\) has the form \(v=\eta c'\) by the same coordinate-plane division, with bounded \(\eta\). A coarse analytic matrix \(A\) with \(Ad'=0\) and \(d'^*A=0\) consequently has the form \(A=c'^*\eta c'\), by applying that division successively to its columns and rows. The estimates use only two fixed Cauchy constants. Apply this to \(B'UB'\) and to \(c_lB_l^{-1}T_lB'\). The latter row before division is \(O(L^{-2}\langle l\rangle^{-s})\). Its squared fine norm includes the factor \(L^2\sum_l\), and hence is \(O(L^{-2})\) after division. These are precisely the bounds used to choose the curl correction with its constant fixed before \(L\). Fourier inversion on any smaller retained strip, and the factors \(d(k_l)\) for fine differences, now give the asserted exponential kernel moments. All reductions of width in this section are made once for the construction; they are not repeated as the depth grows.

One metric and one assignment of each factor

Fix \(L\) before making the following convention. Embed each coarse position at its block anchor, and measure all distances in the same fine-lattice metric. The finite local formulas use a fixed number of primitive kernels. Choose their truncation radii, and the inverse-window radius \(R\), small enough that the sum of the radii in any such local composition is at most \(M/100\). Include block diameters, reference-point displacements, and kinetic-mask radii in this sum. This is possible by reducing the kernel cutoff by a fixed factor depending on \(L\), and then increasing \(H\): the block diameter is fixed and the mask radius is \(O_L(\log H_j)=o(M)\). The omitted-kernel bounds remain \(C_Le^{-c_LM}\) with a possibly smaller \(c_L>0\). Long convolution tails and determinant or inverse chains are not local compositions for this convention: every successive endpoint and window is retained in their records.

For a set \(P\) of seed anchors, let \(N_r(P)\) denote its closed \(r\)-neighborhood. Seeds have their edge, observation, or output-edge types attached. Join two seeds when their \(50M\)-neighborhoods intersect. Write \(P_c\) for the seed set of a resulting component. Distinct sets \(N_{50M}(P_c)\) are disjoint. Consequently each of the assignments below has at most one owner:

The last assignment is applicable because an inverse window differs from the empty geometry only when it sees a frozen site. Objects without an owner remain separate factors. Every primary profile belongs to its own component. Every frozen-site normalization reciprocal belongs to that site’s owner. This assigns each additive row and each multiplicative test once; in particular, the \(8M\) and \(14M\) collections are not additional copies of the \(5M\) collection.

The maximality of \(P_c\) is implemented by compatibility of the \(50M\) footprints in the primary-pattern sum. It is not an additional predicate in the core factor. Testing maximality directly could read a seed \(100M\) away. The pattern is fixed throughout Gaussian integration; it is never recomputed from a Gaussian value.

Profiles and the principal logarithm

Choose once and for all \(\vartheta\in C^\infty(\mathbb R,[0,1])\) equal to one on \((-\infty,0]\) and zero on \([1,\infty)\). Define \[G_e(q)=\vartheta\left( \frac{|d_eq|^2-(t/4)^2}{(t/2)^2-(t/4)^2}\right),\qquad G_Y(q,V)=\vartheta\left( \frac{|V_Y-\widehat m_Y(q)|^2-(Wt)^2}{3(Wt)^2}\right).\] The tag in \(\widehat m_Y\) is retained wherever its fallback branch is used. The factor for a selected primary is \(1-G\); otherwise it is \(G\). These give exactly the sector decomposition by \(G+(1-G)=1\). For every fixed derivative order, a derivative of \(G_e\) has support in \(\{|d_eq|\le t/2\}\), and a derivative of \(1-G_e\) has support in \(\{|d_eq|\ge t/4\}\). The corresponding observation bounds are \(2Wt\) and \(Wt\). Here and below derivatives are taken only on a smooth branch; measurable physical-spin dependencies are held fixed by the transport identity (133).

Every exterior spin is farther than \(3M\) from the seeds. Its block therefore contains no edge or observation primary; the locality convention makes its diameter less than \(M/100\). Lemma 36 applies on the full sector. It also applies locally to every spin of a selected assigned row: that row is farther than \(2.5M\) from every seed, and its block tests lie inside the retained \(14M\) neighborhood.

Here is a completely smooth extension convention. Put \(a_t=2K_Lt\), enlarging \(K_L\) if the finitely many row reference points require it. On \(S^3\), use a smooth function of the geodesic angle that equals one through \(a_t\) and zero from \(2a_t\), multiply the principal logarithm by it, and extend the product by zero. This defines a global smooth frozen-angle function, since \(2a_t<\pi/2\). It equals the actual frozen angle on every selected-row support just described.

On an exterior coordinate use a protector equal to one through \(|\xi|=\pi/2\) and zero from \(3\pi/4\). Also, when that coordinate is read by a core factor or another protected hard factor, attach an alignment profile equal to one through \(a_t\) and zero from \(2a_t\). Both profiles are smooth functions of \(|\xi|^2\) and remain attached. For a compulsory core or mandatory old covering weight they multiply the factor itself. For an optional hard exponential \(e^V\), first open it and prepare instead \[1+\chi_{\rm att}(e^V-1),\] where \(\chi_{\rm att}\) is the product of its attached profiles. Thus the profiles multiply its selected letter, including an old nonremote masked term or a hard kinetic tail. This equals \(e^V\) on the full sector and retains the small bound on \(e^V-1\). Opening \(\chi_{\rm att}e^V\) instead would introduce an unpriced order-one failure outside its plateau, and is not the prescription. The alignment profile is the explicit larger smooth plateau allowed in the preparation; it imposes no hard moving-angle predicate. It is one on the full sector and on the local supports needed for the reserve. Its usefulness at the outer boundary of the retained defaults is that a spin read by a boundary default need not have every test of its block retained. Every such read is nevertheless in a small chart on the support of the attached factor and all its derivatives.

The separate global chart profile is one through \(\pi/4\) and zero from \(\pi/3\). Multiplication by this profile before extending the angular integral to \(\mathbb R^3\) changes no sector integral, by Lemma 36. It removes the other exponential preimages. Its subsequent expansion as one plus a failure factor does not detach the protectors or alignment profiles just specified.

An explicit core factor and its one-time energy charges

Let \(J_c\) be the numerical rows assigned to \(c\), \(J_c^{\rm sel}\) the selected \(\mathcal F,\mathcal B\) rows among them, and \(S_c\) the stationary-square rows assigned to \(c\). Write \(L_i\) for the appropriate row among \(\mathcal F,\mathcal B,\mathcal C,\mathcal U,\mathcal N, \mathcal Z\). Put \[K_c=\sum_{i\in J_c}L_i -\frac12\sum_{i\in J_c^{\rm sel}}|D_i\xi+F_i|^2, \qquad H_c=\frac12\sum_{i\in S_c}|r_{s,i}|^2.\] All rows in the first sum are included with their original sign. In particular no second copy of a direct energy reserve is introduced for a mask witness or a mean-error witness.

Let \(\mathcal P_c\) be the product of the primary profiles of \(c\) and the defaults assigned to \(c\). Let \(\mathcal I_c\) retain the local inventory checks: a marked true bad input edge is supplied once by its old covering label, and every unmarked selected input primary has the factor \(\mathbf1_{r_e\le t}\). Retain each queried output status, including an absent status, in the formula’s data. For an output-empty extension replace these output predicates by the analytic no-flag formula on the stated output graph domain, as in Section 5.2. This is an extension, not a differentiation across the jump of a status indicator.

The leading determinant normalization can be written without an unspecified local compensation. The quadratic acts identically in the three tangent colors. For an exterior site \(x\) define \[h_x=-\frac32\int_0^\infty \left((1+u)^{-1}-(A_{\Omega_x}+u)^{-1}_{xx}\right)\,du,\] and let \(h_\circ(x)\) be its empty-pattern value at the same fine-site phase modulo the blocking lattice. This phase must be retained: \(Q^*Q\) need not be invariant under one-step fine translations. Let \(F_c\) now denote the set of frozen sites of \(c\) (distinct from the affine vector \(F_i\)). With \(\kappa_g=(2\pi g^2)^{3/2}J_{\exp}(0)\), define \[N_c=\kappa_g^{-|F_c|} \exp\left\{\sum_{\substack{x\text{ exterior}\\ x\in N_{3M+R}(P_c)}}(h_x-h_\circ(x)) -\sum_{x\in F_c}h_\circ(x)\right\}.\] The sum includes zero differences harmlessly. Uniform upper and lower spectral bounds on \(A_{\Omega_x}\) give \(|h_x|\le C_L\); at infinity the integrand is \(O_L((1+u)^{-2})\). Thus \(|\log N_c|\le C_LM^2(1+|\log g|)|P_c|\). This is exactly the correction obtained by extracting \(\kappa_g^{|\Lambda|}\exp\{\sum_{x\in\Lambda}h_\circ(x)\}\) while retaining Haar integration at frozen sites. The sum is a fixed scalar per complete \(L\)-block, so it has the required volume form. The residual determinant chains stay separate, and the observation denominator is extracted on every block.

Writing \(\mathcal A_c\) for the product of the attached protectors and alignment profiles, one possible compulsory core factor is therefore \[ B_c=\mathcal P_c\mathcal I_c\mathcal A_c\,N_c \exp\{-b(K_c+H_c)\}. \tag{251}\] The remaining factors are exactly the rows, stationary squares, defaults, Jacobian ratios, and determinant or inverse chains without an owner, and the separately supported old terms. Subtracting every selected affine square once and multiplying back its Gaussian integral is an identity. The assignments, together with the plateau equalities proved above, consequently preserve the full sector integral exactly.

For clarity we give the charging argument for the reserve in this notation. The quantitative inputs are the stack and ledger bounds (119)–(121), with their weighted row and column sums. An input seed with \(t/4\le r\le t\) has a direct square at least \(ct^2\); for \(r>t\) its direct row is at least \(ctr\) once \(L\) is sufficiently large. A noise seed has \(|Y-w|^2/2\ge ct^2\) once \(W\) is large. An output seed has the positive direct \(\mathcal C\) reserve displayed in Section 5.2. The second-stack \(\mathcal C\) contribution cancels and \(\mathcal N\) is nonnegative.

Only a failed input kinetic mask can make a second-stack \(\mathcal F\) contribution negative. Charge that contribution to all its bad-edge witnesses with the following upper bound, so no choice of witness or charge multiplicity is hidden. A witness \(f\) can occur in at most \(C(1+\log(r_f/t))^2\le Cr_f/t\) mask tests. Splitting the convolution into chords at most \(t\) and chords greater than \(t\), and using its column sum for the latter, bounds the total negative contribution by \[\frac{Ct}{L}\sum_{f:r_f>t}r_f.\] Every witness of an assigned row belongs to the same primary component: the mask radius is smaller than \(M/100\) and that witness is itself an input seed. Its direct row is therefore assigned to this component. Choose \(L\) so the displayed loss consumes less than a fixed fraction of those direct reserves.

For the mean term, divide the observation blocks read by assigned rows into blocks all of whose chords are at most \(T\) and their complement. The former cost at most \(C_LtT^2\) per block. In the latter, use \(|m-\widehat m|\le (C/L)\sum_{e\subset Y}r_e\). Chords greater than \(t\) cost another \((Ct/L)\sum_{r_f>t}r_f\). Chords at most \(t\) cost at most \(C_Lt^2\) per such block. Each such block has a witness \(r_f>T\), so their number is bounded, with fixed multiplicity, by \(T^{-1}\sum_{r_f>T}r_f\). This last cost is absorbed by the unused \(ctr_f\) reserve because \(t/T\to0\). This uses the same reserves with specified fractional budgets, rather than paying for a witness twice.

Selected assigned rows are farther than \(2.5M\) from every seed. Their masks are enabled, their means use the smooth branch, and no row in this collection supplies a seed or failed-mask reserve. Their small chart bounds and Taylor’s formula give \[|R_i-R_i^{\rm aff}|\le C_LM^Dt^2, \qquad |R_i|+|R_i^{\rm aff}|\le C_LM^Dt.\] Thus removal of their squares costs \(C_LM^Dt^3\) per row, independently of the preceding charges. There are at most \(C_LM^2|P_c|\) rows, blocks, or local tests in any fixed neighborhood of \(P_c\). This follows by covering that neighborhood by one fixed-radius ball about each seed; overlap reduces, rather than increases, the count. The number \(D\) of polynomial losses is fixed by the finite local formulas and the fixed derivative order, not by the geometry of \(P_c\). Numerical truncation and the remaining small-chord mean terms therefore give \[K_c\ge c t^2|P_c| -C_LM^D|P_c|\{t^3+tT^2+te^{-c_LM}\}.\] Increasing \(H\) absorbs the last line as in the stated scale choices. Since \(H_c\ge0\), (251) then gives \[|B_c|\le \exp\{-c_Lp^2|P_c|+C_LM^D(1+|\log g|)|P_c|\}.\] The same calculation holds on the supports of differentiated retained profiles: primary lower bounds and default upper bounds persist, and the selected angles remain the principal angles by Lemma 36. Alignment profiles additionally control outer-boundary reads. Product-rule differentiation does not discard an undifferentiated test; it replaces it by a derivative with support in the same closed support. Hard physical-spin tests are transported, not differentiated. The no-output-flag extension uses the positive-mask, unclipped formula through \(4t'\); it changes only the fixed constants in the chart and Taylor estimates.

The read sets of the prepared factors

The following conventions specify complete read sets. A raw position means every endpoint of a row stencil, graph, or recorded path, rather than only its anchor. Add the mean window and output reference for each exterior spin read. Add both centered inverse windows for each covariance entry used. Add the \(3M\) seed-status neighborhood needed to decide membership in every such window. A status read includes both possible answers and is retained when the factor is considered alone.

A noncore factor retains \(N_{20M}\) of every raw position and its joining record. The \(14M\) remote query, \(3M\) hole query, and finite local composition enlargement fit in \(18M\) under the chosen convention. For a core all raw local tests lie in \(N_{14M}(P_c)\); their enlargement fits in \(N_{18M}(P_c)\subset N_{50M}(P_c)\). Old nonremote factors, long tails, and long chains keep their own complete supports, and meet a core or other factor by support intersection. They are not absorbed into a fixed-radius core while forgetting their distant endpoints.

Exact local marginals

Let \(E(P)=\Lambda\setminus N_{3M}(P)\) and write \(A_0\) for the empty-pattern quadratic. Every row incident on an exterior column is selected: its anchor is farther than \(3M-M/100>2.5M\) from the seeds. Consequently \[A=A_0[E(P),E(P)].\] There is no additional selected-row dependence in this principal matrix. For \(x\in E(P)\) let \(\Omega_x=E(P)\cap B_R(x)\) and write \[\Pi_P(y,x)=\mathbf1_{y\in\Omega_x} \bigl(A_0[\Omega_x,\Omega_x]^{-1}\bigr)_{yx}, \qquad C_P=\tfrac12(\Pi_P+\Pi_P^*).\] The positivity of \(C_P\) is supplied by the window estimate in Section 5.2.

Finally, at a fixed total derivative order only a fixed number of factors are differentiated. The number of placements is polynomial in the number of read positions, hence in the total support load times a fixed power of \(M\). For completeness the vector field used in protected transport is, on each protected coordinate, \[W_x=-g^{-1}(d\exp_{\xi_x})^{-1} \bigl((\partial_vq_x)T_x^{-1}\bigr), \qquad \xi_x=\mu_x+gZ_x.\] Here \(\partial_v\) holds \(Z\) fixed, and the inverse maps the tangent space at \(\exp\xi_x\) to the imaginary quaternions. Extend this field smoothly from the compact protector chart and set its other components to zero. All fixed-order derivatives of this inverse differential are bounded on that compact chart. The field derivatives of \(\mu\) come from the displayed window formula: its inverse matrix is independent of the output field at fixed primary statuses, and the derivatives of its affine arguments use only the smooth finite-row and frozen-angle formulas. Their cutoffs cost fixed powers of \(t^{-1}\), paired with normalized directions of size \(t\) times the prescribed distance weights. Window and free-history derivative or difference bounds supply the corresponding parameter estimates. These facts, and the stated prediction bounds for remote rows, give fixed polynomial bounds for \(W\), its divergence, and their iterated derivatives in \(g^{-1},M,p\) and the number of reads.

At fixed geometry \(C_P\) has no output-field dependence. For a parameter or history comparison interpolate its two positive marginal matrices linearly. Uniform ellipticity is preserved, and the covariance score is \[\frac12\left(Z[I]^*C[I,I]^{-1}\dot C[I,I] C[I,I]^{-1}Z[I] -\operatorname{Tr}(C[I,I]^{-1}\dot C[I,I])\right).\] The window difference estimate bounds \(\dot C\) with its own discrepancy factor. The trace costs at most a constant times \(|I|\), and the other term is a Gaussian polynomial of fixed degree. Thus the derivative scores have the polynomial load dependence needed for the joint estimate, with the discrepancy retained. Every hard nonremote factor still meets a retained core, an old covering weight carries a primary inventory edge, and a voluntarily hard tail retains its exceptional chain estimate. This identifies the reserve used by the joint-factor estimate; it does not replace that estimate by a claim of independence for overlapping supports.

Verification for the eight classes of prepared factors

We apply Lemma 38 on the union of the read coordinates of a finite selection in one fixed primary pattern. The scalar-coordinate covariance bound follows from (123); it is uniform in the pattern, its exterior, and the admitted period. Empty exterior means a zero-dimensional Gaussian and causes no exception. The complete read sets and attached profile rules are those of Appendix 14.

For the eight classes listed in Section 5.3, the following are the required envelopes and reserves.

1. The unassigned nonlinear rows have deterministic envelopes \(C_Lg\operatorname{poly}(M,p)\). Their normalized smooth derivatives retain that order: a cutoff derivative contributes \(t^{-1}\) together with a direction of size \(t\) times its prescribed distance weight.

2. The stationary squares have order \(g^2\); the \(\mathcal N,\mathcal Z\) remainders have exponential accuracy in \(M\). Their fixed-order derivatives have the same envelopes by (127) and the truncated free-kernel estimates.

3. A remote canonical term retains (129), and a remote regular error retains its actual derivative norm. A nonremote term is a selected hard letter with the attached profiles prescribed in Appendix 14; it meets a compulsory core by its recorded nonremote test.

4. A kinetic tail retains every path and mask query. If it is smooth its exponential envelope suffices. If it is treated as a hard letter, that same exceptional envelope pays its fixed-order transport loss. The preparation is \(1+\chi_{\rm att}(e^V-1)\), so it introduces no unpriced cutoff failure.

5. The ratio-chain letters use (130). Their scalar endpoint counts are polynomial in their recorded loads and in \(M\). To sum their eighth roots, fix a support exponent \(B>0\), a support hit, and a number \(r\ge1\) of residual hops. The finite-range convention of Section 14.1 gives \(O(M)\) range for each matrix hop and window. There are at most \(C_LM^D\) choices at each hop; choosing which of the \(O(r)\) retained windows has its fixed \(O(M)\) support neighborhood containing the hit costs only a further polynomial. Thus the number \(N_r\) of literal paths is at most \[N_r\le\operatorname{poly}(M,r)(C_LM^D)^{r+O(1)}.\] For a resolvent chain, first integrate its coefficient in the resolvent parameter, using the integrable \((1+u)^{-2}\) bound in Section 5.2. The envelopes \(e_\alpha\) then index discrete paths. Apply (126) with support exponent \(8B\) and Hölder’s inequality for this finite path set to obtain \[\begin{aligned} \sum_{\alpha:\,r\text{ hops}}e^{Bs_\alpha}e_\alpha^{1/8} &\le N_r^{7/8} \left(\sum_{\alpha:\,r\text{ hops}} e^{8Bs_\alpha}e_\alpha\right)^{1/8}\\ &\le\operatorname{poly}(M,r) \exp\{-c_LMr/8+O_L(r\log M)+CBr\}. \end{aligned}\] This is summable over \(r\) and tends to zero as \(H\) increases. Since the envelopes are less than one, their square-root sum has the same upper bound. Increasing \(H\) makes this support-hit sum less than \((64c_*)^{-1}\). The exponential decrease in \(M\) times the number of hops dominates the polynomial endpoint count, proving (253) as well.

6. The Haar-Jacobian letter is smooth and has order \(g^2\) with its normalized derivatives, since the Jacobian has zero linear term.

7. The opened edge, observation and chart defaults are the failure factors. Equation (132) supplies the simultaneous bound with \(\tau=c_Lp^2/M^D\). It holds on all fixed-order derivative supports, by the prediction estimates and the retained support properties of Section 5.3.

8. An old covering weight retains its sup envelope. Its nonempty inventory supplies a marked primary input edge, hence a compulsory core. Its measurable physical-spin arguments are fixed by the common transport; the weight is never field-differentiated.

The compulsory core itself is (251). Its reserve, proved there on the retained profile and derivative supports, gives a deterministic envelope \(e^{-\kappa_Lp^2s_c}\) after increasing \(H\). Normalized derivatives of its smooth factors or exponent cost only a fixed polynomial in \(g^{-1},M,p\) and the total number of reads. The same reserve remains during interpolation of two core exponents at fixed physical spins, because both endpoints obey it.

For a smooth product without a hard factor, the ordinary derivative bounds already stated preserve the ordinary orders. In a product with hard physical-spin dependencies, use the single simultaneous transport at the end of Appendix 14. That calculation supplies the derivatives of the mean, covariance score, vector field and divergence required above. There are at most a polynomial number of placements at a fixed total derivative order. The \(L^4\) norm of each score is bounded by a fixed polynomial of the number of reads and by fixed powers of \(g^{-1},M,p\). Fixed powers of \(M,p\) can be absorbed in a further fixed power of \(g^{-1}\) for sufficiently large \(H\). Every use that requires this loss has a core, failure or exceptional ratio/tail reserve by the eight cases just listed. A marked coupling, history or input discrepancy retains its own factor in these same bounds, using the window difference identity and the interpolation at fixed physical-spin arguments. For an input discrepancy, telescope a finite product into terms with one marked factor \(F_{\alpha,2}-F_{\alpha,1}\). Its coefficient, regular-error or covering-weight difference supplies the replacement envelope in Corollary 39; the other factors use envelopes uniform for the two inputs. The marked support retains the same core or exceptional reserve whenever transport requires it.

Corollary 39 therefore proves (131). The support-hit sums of the resulting majorants are obtained exactly as at the end of the proof of Lemma 11: projected old records use (109), canonical paths use their fixed exponential weight, core patterns are paid by their \(p^2\) reserve, and chains use (126). Reserve a further fixed positive amount of support exponent for the polynomial in total load. The conditions (148) leave this reserve while retaining \(e^{-p^{3/10}}\) for output-seeded components. This is a product estimate under one Gaussian, not an independence assertion about overlapping factors. Exact independence is used only for disjoint complete supports, where Lemma 37 proves it.