Proof. We use the bridge-decomposition method of Hammersley and Welsh [13]; see also Madras and Slade [16] and Duminil-Copin and Hammond [6]. We give the decomposition with the present honeycomb endpoint conventions, using the positive strip bound to control each weak bridge.

Submultiplicativity gives existence of \(\mu\). Split a walk at a vertex of minimum height, choosing the first such vertex in a fixed traversal order when necessary. Its two tails start at that vertex and stay above its height. Translate this vertex to one of two fixed representatives, according to its triangular-band type. Recording the two tails and the split time recovers the original walk, because its original root is fixed.

Decompose each nonempty tail by stopping at its last global maximum, then at the last minimum of the remainder, and continuing alternately. Every piece is a weak bridge: all its vertices lie between the heights of its endpoints. There are no horizontal edges. Its positive spans, measured in thirds of a triangular band, are integers, are nonincreasing, and are strictly decreasing after a possible equality of the first two. Their sum is at most \(2n\). Consequently there are \(O(\sqrt n)\) pieces and at most \(\exp(O(\sqrt n\log(n+1)))\) possible span lists, even after recording the split time and the two initial types.

For a fixed initial vertex and a specified directed span, the total \(\rho^{\#\text{edges}}\)-weight of weak bridges is bounded by an absolute constant. To obtain this from the strip bound \(\mathcal B_H\le1\), choose horizontal band boundaries immediately beyond the two endpoint heights. At a lower endpoint of lower-center type append the vertical half-edge to the lower boundary. At the other type first append the fixed descending edge, then the vertical half-edge. Make the reflected construction at the upper endpoint. All added vertices are outside the old height range, so the extension preserves self-avoidance and has bounded multiplicity. The starting boundary port and strip height are determined by the starting vertex and span. The extension adds at most three vertex-weight factors relative to the old edge count. The same argument applies to downward bridges by reflection.

For each span list, ignore avoidance between successive pieces and sum their weights successively. This gives a constant to the power \(O(\sqrt n)\), and hence \[c_n\rho^n\le \exp(O(\sqrt n\log(n+1))).\] Taking \(n\)th roots yields \(\rho\le\mu^{-1}\). If the inequality were strict, choose \(r\) with \(\rho<r<\mu^{-1}\). The unrestricted walk series at \(r\) converges, so its tail at \(\rho\) decays exponentially in length. A bridge crossing \(N\) bands has length at least a positive constant times \(N\), up to endpoint constants. Its mass would therefore decay exponentially in \(N\), contradicting \(\mathcal B_N\asymp N^{-1/4}\) from the strip calculation. ◻

Proof. A cylinder path lifts to a planar self-avoiding path and uses at most \(O(NM)\) vertices. Proposition 2 therefore bounds \(b_M(i,j)\) and all same-side path sums by \(\exp(o(M))\), with \(N\) fixed. A finite transfer on partial matching states, with one unmatched port attached to the source and every closed cycle discarded, computes \(b_M(i,j)\). Such connectivity records all necessary information: distinct columns have disjoint vertices. Hence \(\sum_Mb_M(i,j)z^M\) is rational.

In fact \(b_M(i,j)\) decays exponentially in \(M\). Otherwise, because its coefficients are nonnegative and subexponential and its generating function is rational, some entry has \(\sum_Mb_M(i,j)=\infty\). We construct a projection-invalid connector for these precise endpoint types. A single column in the planar lift is an infinite chain of rhombi, with row labels periodic modulo \(N\). Starting at its west port of type \(j\), take the single west-to-north turn, then the single south-to-north connection through successive rhombi. Choose an integer \(k\) with \(i+kN>j+2N\) and take the south-to-east turn in row \(i+kN\). The three kinds of one-arc tile used here have strictly positive weights at both physical parameters \(\lambda,2\lambda\); only one arc is placed in each tile. Since different rhombi have disjoint interiors, the lifted path is self-avoiding and meets the column’s west and east cuts only at its endpoints. After projection, however, its intermediate south-to-north route repeats vertices in a row visited one period earlier. This is a width-one connector from type \(j\) to type \(i\) with some positive weight \(a\), depending only on the fixed cylinder.

Concatenate lifts of bridges counted by the divergent entry with copies of this connector, always in successive disjoint column slabs. Their interiors are disjoint, so the resulting planar path is self-avoiding. To recover the concatenation, list all column cuts that the path meets exactly once. Between two adjacent column cuts in this list, a width-one subbridge inside any original valid slab is still projection-valid. The inserted connectors are precisely the projection-invalid width-one subbridges: they occupy one entire column and their two cuts are crossed only at their endpoints. Thus the connectors, the intervening slabs, and their original paths are uniquely recovered from the concatenated path.

Choose finitely many valid slab widths whose total mass is \(B\) with \(aB>1\). The width generating function of one valid-slab–connector block is \(g(z)=a\sum_M b_M(i,j)z^{M+1}\), with the sum restricted to those widths. It satisfies \(g(0)=0\) and \(g(1)>1\). Hence the generating function \(\sum_{r\ge0}g(z)^r\) of recoverable concatenations has radius strictly less than one. Each slab has at most a fixed constant times its width in vertices, because its projection is self-avoiding on a fixed cylinder; the connector has bounded length. Consequently all these planar paths of total width \(w\) have length at most \(C_Nw\). Their total critical weight is bounded above by the sum of \(c_n\rho^n\) for \(n\le C_Nw\), up to fixed endpoint factors. This is \(\exp(o(w))\) by Proposition 2, contradicting the radius just obtained. ◻

Proof. A nonzero diagram propagated from a nonempty input pairing either has a west-to-east spanning path or closes a discarded cycle. The number of path segments is bounded at fixed \(N\). Discarding their mutual avoidance bounds their weights by a product of an exponentially decaying spanning sum from Lemma 6 and subexponential same-side sums. Therefore the transfer on the subspace of zero empty component has spectral radius strictly below one. The empty component itself is invariant. Writing the transfer in these two blocks proves uniqueness and local analyticity of its normalized fixed vector, and convergence from a vacant far boundary proves the half-cylinder interpretation. The rational stationary vector \(V_N/h_N\) agrees with it wherever defined nearby, hence extends to the physical parameters. Horizontal reflection gives the stated right-half arguments. Vertical reflection also recovers the covariance proved algebraically above. ◻

Proof. The right-half arguments are \(q^3/x_i\). Equation (4), homogeneity, and the per-slot degree bounds for \(V_N\) give the polynomial bound for \(H\): occupied half-powers occur in pairs in the contraction. To exchange sites within a group, move a symmetric braid box from one side of the contraction to the other by transposition. The two states braid in opposite directions because the starred arguments reverse differences. Loop parity is preserved: following the arcs in the box layer shows that each resulting loop has an even total number of \(X\)-hits on the two interfaces of that layer. Thus any cycle discarded on one side has zero gluing weight as well. This proves symmetry and also permits a same-group splitting layer to be moved through the contraction. At the descending specialization on the starred side, apply \(R(j\lambda)=S_jS_j^{\mathsf t}\) and the ascending splitting identity. To cancel \(S_j\), project its output onto diagrams whose first new slot is vacant. For \(S_2\), omitting that slot leaves the diagonal map with coefficients \(1\) on a vacancy and \(\rho\) on an occupied strand. For \(S_3\), both new slots are then vacant; omitting them gives the identity. Thus \(S_j\) is injective, and \(S_jU_{\rm red}=S_jS_j^{\mathsf t}U_{\rm desc}\) implies \(S_j^{\mathsf t}U_{\rm desc}=U_{\rm red}\). This proves the reduction of \(Z\).

Cyclic covariance and the last-slot leading coefficient show that sending one argument to infinity inserts a vacant site in the left vacuum. Reflection gives the same statement in the right vacuum. Horizontal reflection gives the limit at zero.

For completeness, the simultaneous two-site limit requires more than the individual limits. Reorder within groups so that \(x,y\) are adjacent across a marker, and rotate them to the last two positions. Write \(P\) for the other \(N-2\) arguments. We claim \[ \frac{V_N(P,x,y)}{(xy)^{N-2}} =V_{N-2}(P)\otimes V_2(x,y)+o(|x|), \qquad V_2(x,y)=(x+y)\varnothing+2c\sqrt{xy}\,\mathrm{cup}. \tag{6}\] Fix square-root branches on the scaling ray. In the interpolation in \(y\), the deletion nodes are \(y_z=q^3z\) for \(z\in(P,x)\). For \(N\ge3\), nodes belonging to \(P\) have data \(O(|x|^{N-1})\), cardinal factors \(O(|x|^{N-3})\), and additional occupied square-root ratios at most \(O(|x|^{1/2})\). After division by \((xy)^{N-2}\) they are \(o(|x|)\). The leading vacant interpolation term becomes \((y-y_x)V_{N-2}\otimes\varnothing+o(|x|)\). At \(y_x=q^3x\) the data are \[(x+y_x)\prod_{p\in P}(p-q^{-3}x)(p-q^6x)\,S_3V_{N-2}(P).\] The cardinal factor is asymptotic to \((y/y_x)^{N-2}\), with an extra \(\sqrt{y/y_x}\) on occupied output. Since \(q^{-3}q^6=q^3\), these terms give exactly the remaining terms in Equation (6). The expression for \(V_2\) follows directly from the size-two interpolation. Division by the empty components gives factorization of the normalized left vacuum, and reflection gives that of the right vacuum. The intersection-parity rule factors on the two subsystems, proving both the pair limit and the displayed \(Z_{x,y}\).

To prove uniqueness, the zero-site value and omission already settle \(N=1\). At \(N=2\) with one site in each group, omission at zero and infinity leaves only a possible difference \(Cxy\) of the cleared numerators. Its quotient by \(h_2^2=(x+y)^2\) has a nonzero paired scaling limit unless \(C=0\), so the specified pair limit settles this case. For all other group sizes, induct on \(N\) and subtract two candidate cleared polynomials. The known scalar reductions of \(h_N\) turn the reductions of \(Z\) into identical values of these polynomials at each split. Omission at zero kills their constant coefficient, and omission at infinity kills their leading coefficient. In a variable of a group of size \(s\), symmetry and the two reductions supply \(4(s-1)\) roots, namely \(q^{\pm2}\) and \(q^{\pm3}\) times every other same-group argument. If \(s>b\), these roots, together with the zero constant coefficient, exceed the remaining degree \(2N-3\). If \(s=b\ge2\), the difference must be a constant multiple of \[ \left(\prod_i x_i\right) \prod_{\substack{i<j\\\text{same group}}} (x_i^2+x_j^2)D_0(x_i,x_j). \tag{7}\] Indeed its degree in each variable is \(1+4(s-1)=2N-3\), exactly the remaining bound, so the quotient has degree zero in every variable. When one argument from each group tends to infinity, this factor has the same leading order as \(h_N^2\) and a nonzero generic coefficient. The pair limit forces the rational difference to tend to zero, so its constant multiplier vanishes. The size-zero and size-two conditions cover the remaining balanced cases. ◻

Proof. Integrability and contour geometry. All coefficient terms on the right have exactly \(N\) elementary particles, because \((s-M)+(b+M)=N\). A particle at real coordinate \(u\) receives decay \(O(e^{-N|u|})\) from the \(N\) sites; a pair interaction grows at most \(O(e^{|u-u'|})\). The inequality \(|u-u'|\le|u|+|u'|\) therefore bounds the full integrand away from its prescribed poles by a constant times \(\exp(-\sum|u|)\), counting elementary coordinates with multiplicity. The same estimate applies to smaller elementary degree and to fused particles or residues. Constants may depend on \(N\), sites, and the fixed contours. This proves the convergence needed for contour movements and removal of their vertical ends.

For \(\varepsilon<1\), temporarily move the far singletons to a common shift \(h\) and the far doubles to \(h-1,h\), where \[3<h<\frac{7-\varepsilon}{2}.\] The offsets are measured outward from each group’s site ray, in units of \(i\sigma\beta\): \[\begin{array}{c|c|c} &\text{fixed offsets}&\text{temporary offsets}\\\hline \text{main singleton}&0&0\ \mathrel{% \BeginAccSupp{method=hex,unicode,ActualText=27F6}% \longrightarrow% \EndAccSupp{}% }\ h\\ \text{far singleton}&11/4&h\\ \text{far double}&(5/2,7/2)&(h-1,h)\\ \text{site residues}&\text{none initially}&1\ \text{or}\ (1,2). \end{array}\] A parenthesized pair shares one real coordinate. Only the far contours are moved in this first step; the main-contour arrow is the subsequent deformation. Moving the far contours crosses no pole. Indeed the interaction of a double with any particle has no poles after its two constituents are multiplied. Relative to the double midpoint, the same-group fused \(E_sE_s\) has only positive exponents at shifts \(\pm1/2,\pm5/2\); the cross-group product has exponent two at \(-1/2,5/2\). The same observation applies to pairs of doubles. Far particle–site poles are avoided because all far constituent shifts stay in \((2,7/2]\). For opposite-group singletons the cross difference lies strictly between zero and seven in shift units.

Residues and cancellation of unbound particles. Move the main contours from shift zero to shift \(h\). A same-group site can collect an ordered string of length \(a=0,1,2\), at its own coordinate plus \(i\sigma\beta,\ldots,i\sigma a\beta\). The first attachment occurs at shift one; interaction with it permits a second at shift two. An attempted third at shift three is canceled by the zero in its interaction with the first. Quadratic coincidence zeros likewise cancel attempts to attach twice at the same root. A main particle can also attach one unit below an original far singleton, converting it into a double. These are all possibilities: fused doubles have no particle poles, and \(2h+\varepsilon<7\) excludes cross-group poles.

Here is an ordering of the deformation that avoids any ambiguity from moving poles. Divide \([0,h]\) into generic layers of thickness less than \(1/3\), separating the levels \(2\) and \(h-1\). In each layer move the remaining unbound variables one at a time; freeze every picked residue immediately, including during the rest of that layer. Unbound variables in a common group then never differ by a full unit of shift, so they do not create poles with one another. For fixed generic distinct real coordinates of original far particles, also distinct from site offsets, these are ordinary one-variable contour moves across simple poles. An attached far spectator appears as the complete double interaction. An unconverted far singleton is more than one shift unit beyond every site-bound root. Thus residual integrands have no unaccounted coincident-spectator singularity and satisfy the preceding decay bound. The identities can consequently be integrated over the original far spectators as well.

The internal weight of a string, including its factors with its own site and excluding activities, is \[ r_0=1,\qquad r_1=\frac1{2d},\qquad r_2=\frac{c^2}{d^2}. \tag{16}\] With measure \(du/(2\pi)\), the oriented residue multiplier is \(i\sigma\). The first \(F_s\) pole contributes \(1/(2d)\). The second attachment contributes \(-4c^2\) from the \(E_s\) pole at relative \(i\sigma\beta\), and \(F_s(2i\sigma\beta)=-1/(2d)\) from its site factor. Multiplication gives \(r_2\). Attachment at relative \(-i\sigma\beta\), below a far singleton, instead contributes \(4c^2\).

Fix a site-string pattern. The effective far singleton activity is \(-p_\sigma+p_\sigma=0\), from original and unpicked main singletons. The effective far double activity is \(4c^2p_\sigma^2+4c^2(-p_\sigma)p_\sigma=0\), from original doubles and attached original singletons. These cancellations respect all factorials. Explicitly, start in one group with \(n,t,D\) particles of the three species. If \(r\) main particles fill the specified site strings and \(a\) attach to \(a\) of the original far targets, summing labeled assignments leaves factorial coefficient \[\frac1{(n-r-a)!\,(t-a)!\,a!\,D!}.\] Each assignment determines exactly one contour term through its choices at shifts \(1,2,h-1\). The remaining two kinds of singleton and two kinds of double can therefore be combined by the binomial formula. This argument is coefficientwise in \(p_+,p_-\), so all canceling terms have the same elementary total. Only completely site-bound configurations survive. In particular every cancellation used for the degree-\(N\) coefficients takes place inside the finite truncation defining \(I\).

Identification with the finite label formula. Assign label \(l=\sigma(1-a)\) to a site in group \(\sigma\) with string length \(a\). The coefficient constraints become \(\sum_{X}l_i=\sum_{Y}l_i=M\). We check their mutual weights explicitly. Use the formal string weights and their pair-phase identity (13) established before the finite formula. To multiply these phases without a size restriction, put \(\sigma_i=1\) on \(X\), \(\sigma_i=-1\) on \(Y\), and \(v_i=2+l_i^2\). Neutrality gives \(\sum_i\sigma_i l_i=0\), while \(l_i^3=l_i\). Hence \[\begin{aligned} \frac12\sum_{i<j}\sigma_i\sigma_j(v_i l_j+v_j l_i) &=\frac12\left[ \left(\sum_i\sigma_i v_i\right) \left(\sum_i\sigma_i l_i\right)-\sum_i v_i l_i\right]\\ &=-\frac32\sum_i l_i=-3M. \end{aligned}\] The cross sign is \((-1)^{sb-(\sum_Xl_i)(\sum_Yl_i)}=(-1)^{sb-M^2}=(-1)^{sb+M}\). Thus the product of all pair phases is exactly \(q^{-3M}(-1)^{sb+M}\). Finally, since \(\sum_i a_i^{\mathrm{len}}=N\) for the string lengths \(a_i^{\mathrm{len}}\), \[(2d)^N\prod_i r_{a_i^{\mathrm{len}}}=(-2c)^{\sum_i l_i^2}.\] Indeed the per-site ratio of the right-hand weight to \((2d)^a r_a\) is \((-2c)^{1-a}\), whose product is one. Taking \(\tau=q^3e^{i\eta}\) in Theorem 9 now proves Equation (15).

Confluent physical limits. Return the far contours to the fixed shifts in the definition of \(I\). Their fused pair products have no pole at \(\varepsilon=1\). On these fixed contours the same decay bound persists in a neighborhood of that value and of confluent real site offsets. This justifies both final limits. ◻

Proof. For real \(r\), \(P_j(-r)=-\overline{P_j(r)}\). Each fused product has even total exponent, which gives the asserted conjugation after a real sign has been extracted. Real zeros of an elementary factor occur only at \(r=0\) and \(j\in8\mathbb Z\). Combine the exponents before testing for a zero, in particular for the two constituents of a double. Ordered by main singleton, far singleton, and far double, the extracted pair signs and vanishing orders are \[\begin{array}{c|c|c} &\text{sign matrix}&\text{order matrix at }r=0\\\hline\noalign{\vskip 3pt} \text{same group}& \begin{pmatrix}+&-&+\\-&+&+\\+&+&+\end{pmatrix}&2I_3\\\noalign{\vskip 3pt} \text{different groups}& \begin{pmatrix}-&-&+\\-&-&+\\+&+&+\end{pmatrix}&0. \end{array}\] The site–particle signs are \((+,-,+)\) in the same group and \((-,-,+)\) in different groups, with zero vanishing order in both cases. These tables follow directly from \(P_j(0)=-2i\sin(j\beta/2)\) and \(P_{8k}(r)=-(-1)^kr+O(r^3)\): shift the exponents of \(P_0^2P_2P_{-2}/(P_1P_{-1})\) or \(P_0^2P_2^2/P_1^2\) by every constituent difference, add them, and cancel opposite powers. Thus there are no other real singularities.

Let \(n_\sigma,t_\sigma,D_\sigma\) count main singletons, far singletons, and far doubles. Their elementary totals are \(K_+=s-M\) and \(K_-=b+M\), with \(K_\sigma=n_\sigma+t_\sigma+2D_\sigma\). The cross-group site and pair signs in the table have parity \[bK_++sK_-+K_+K_-\equiv sb+M\pmod2.\] This cancels the prefactor in Equation (15). The remaining negative factors are a same-group main–far-single pair, a same-group far-single–site pair, and each far-single activity. Their total parity is \[(n_++s+1)t_++(n_-+b+1)t_- \equiv M(t_++t_-)\equiv Mf\pmod2,\] because \(t_\sigma^2\equiv t_\sigma\) and \(f=t_++t_-+2D_++2D_-\). Doubles contribute only positive signs in the tables. Any fixed extraction of the site–site signs is independent of \(M\). This proves the required imbalance phase. ◻

Proof. First take distinct sites. Write \[K(u)=\frac{\sinh u}{\cosh u+d},\qquad K_0(u)=\tanh\frac{\gamma u}{2}.\] Equation (3) and the definition of \(H_*\) give \[ \frac{\widetilde h_N^2}{\prod_{i<j}H_*(a_i-a_j)^2} =\frac{\det[K(a_i-a_j)]}{\det[K_0(a_i-a_j)]}. \tag{18}\] Here we used the Pfaffian identity \[\operatorname{Pf}\left[\frac{v_i-v_j}{v_i+v_j}\right] =\prod_{i<j}\frac{v_i-v_j}{v_i+v_j},\qquad v_i=e^{\gamma a_i}.\] To prove it, clear denominators: the numerator is an alternating polynomial of Vandermonde degree and is therefore a constant times the Vandermonde. The residue at \(v_{N-1}=-v_N\), followed by induction from size two, determines the constant to be one.

For an even-dimensional skew matrix \(A\), adding the all-ones matrix does not change its determinant. Indeed the rank-one determinant formula gives the extra term \(\boldsymbol1^{\mathsf t} \operatorname{adj}(A)\boldsymbol1\), which is zero since \(\operatorname{adj}(A)\) is skew. Apply this to both matrices in Equation (18), then conjugate by the diagonal factors \(e^{-\gamma a_i/2}\). The resulting reference kernel is \[G_0(u)=e^{-\gamma u/2}(1+K_0(u)) =\operatorname{sech}(\gamma u/2),\] and the actual kernel is \(G(u)=e^{-\gamma u/2}(1+K(u))\). With Fourier convention \(\widehat f(k)=\int_\mathbb Re^{-iku}f(u)\,du\) and \(\zeta=\gamma/2+ik\), their transform ratio is \[ \frac{\widehat G(k)}{\widehat G_0(k)} =\gamma\frac{\cos(\beta\zeta)\sin(\pi\zeta/\gamma)} {\sin(\pi\zeta)} =\frac\gamma2(1+R(k)),\qquad R(k)=\frac1{2\cos(2\beta\zeta)}. \tag{19}\] For the first equality, put \(t=e^u\) and use \[1+K(u)=\frac{t}{t+e^{i\beta}}+\frac{t}{t+e^{-i\beta}},\qquad \int_0^\infty\frac{t^{p-1}}{1+t}\,dt=\frac\pi{\sin\pi p}.\] The beta integral follows by a keyhole contour, initially for \(0<\Re p<1\). The second equality in Equation (19) follows from \(\pi/\gamma=3\beta\) and \(\pi=4\beta\) by the triple-angle identity.

The determinant ratio is now explicit, but direct differentiation at coalescing sites would obscure its size. We instead view the reference matrix as a Gram matrix. Its orthogonal projection will occupy an interval of length \(2\gamma^{-1}\log N\) in the real direction; the trace density on that interval will give the required logarithmic term.

We turn the determinant ratio into a finite-dimensional compression. Let \(\mathcal H\) be the Hilbert space with norm \[\|f\|_{\mathcal H}^2 =\int_\mathbb R|\widehat f(k)|^2v(k)\,\frac{dk}{2\pi},\qquad v(k)=\frac1{\widehat G_0(k)} =\frac\gamma{2\pi}\cosh\frac{\pi k}{\gamma}.\] Its functions have bounded evaluation at every \(z\) with \(|\Im z|<\pi/(2\gamma)\). Write \(k_z\) for the evaluation vector, and \(P=P_N\) for the orthogonal projection onto the span of evaluations and their derivatives of orders \(0,\ldots,m-1\) at \(z=\pm i\beta/2\). Centering the two site rays changes no differences. The derivative evaluation vectors are linearly independent. Indeed, a linear relation among them, written in Fourier variables and multiplied by the common nonzero weight, has the form \(p_+(k)e^{\beta k/2}+p_-(k)e^{-\beta k/2}=0\) on \(\mathbb R\), where \(p_+,p_-\) have degree at most \(m-1\). Sending \(k\) to \(+\infty\) gives \(p_+=0\), and then \(p_-=0\). Their Gram matrix is consequently positive definite.

Start with distinct paired real offsets and permute columns by complex conjugation; the reference matrix becomes the Gram matrix of these evaluation vectors. The actual matrix represents the sesquilinear form with the Fourier multiplier \(\frac\gamma2(I+R)\) inserted. Passing to divided differences gives the derivative vectors at confluence. Hence \[ \frac{\widetilde h_N^2} {(\gamma/2)^N\prod_{i<j}H_*(a_i-a_j)^2} =\det\left(I+PRP\big|_{\operatorname{ran}P}\right). \tag{20}\] The same column permutation is used in numerator and denominator, so it introduces no sign in their ratio. The reference Gram matrix is nondegenerate by the preceding independence argument. Moreover \(\operatorname{Re}R(k)\ge0\) for real \(k\): since \(2\beta\operatorname{Re}\zeta=\pi/3\), the real part of \(\cos(2\beta\zeta)\) is \(\tfrac12\cosh(2\beta k)>0\). Thus \(I+PRP\) is strictly accretive and invertible. Since the removable factors \(H_*(0)\) and \(H_*(i\beta)\) are nonzero, the compression identity already proves \(h_N\ne0\) at these balanced confluent sites. The rest of the proof determines its logarithmic size.

The projection has a useful exact localization formula: \[ \|(I-P)k_z\|^2=\|k_z\|^2 \prod_{\pm} \left|\frac{e^{\gamma z}-e^{\pm i\gamma\beta/2}} {e^{\gamma z}+\overline{e^{\pm i\gamma\beta/2}}}\right|^{2m}. \tag{21}\] To see this, augment the reference Gram matrix by \(k_z\) and take its Schur complement. Under \(w=e^{\gamma z}\), its kernel is \(1/(w+\overline{w'})\) up to separate one-point factors. The Cauchy determinant formula follows by clearing denominators and using antisymmetry, with the constant fixed by residues. Its ratio after augmentation is exactly Equation (21). Divided differences give its confluent version and prove the nondegeneracy of the derivative Gram matrix.

For each fixed horizontal line in the evaluation strip, this implies \[ \|Pk_z\|^2\le C\min\{1,Ne^{-\gamma|\Re z|}\}. \tag{22}\] Indeed, for \(a=e^{\pm i\gamma\beta/2}\) and \(w=e^{\gamma z}\), \[1-\frac{|w-a|^2}{|w+\overline a|^2} =\frac{4\Re w\Re a}{|w+\overline a|^2} \asymp e^{-\gamma|\Re z|}.\] The product in Equation (21) is therefore within \(CNe^{-\gamma|\Re z|}\) of one, and is at most \(\exp(-cNe^{-\gamma|\Re z|})\). In particular, translated to the frame of any real \(x\) with \(|x|\le(1-\delta)L\), the projections \(P_N\) converge strongly to the identity, uniformly for fixed \(\delta>0\). The complement tends uniformly to zero on every fixed evaluation vector, and these vectors span a dense subspace by Fourier uniqueness; contraction of the projections extends this to all of \(\mathcal H\).

Integrating Equation (22) on horizontal lines near the two edges of the evaluation strip gives, for every fixed \(\epsilon>0\), \[ \operatorname{Tr}(Pe^{-\epsilon|k|}P)\le C_\epsilon L. \tag{23}\] Here multipliers are written as functions of \(k\). More explicitly, Plancherel on the line of height \(y\) gives the multiplier \(v^{-1}e^{-2ky}\), and its trace against \(P\) is \(\int_\mathbb R\|Pk_{x+iy}\|^2\,dx\). Choose \(y\) near each strip edge. The sum of the two resulting positive multipliers dominates a constant multiple of \(e^{-\epsilon|k|}\), while each integral is \(O(L)\) by Equation (22).

We claim, for each integer \(p\ge1\), \[ \lim_{N\to\infty}\frac{\operatorname{Tr}((PRP)^p)}{2L} =\int_\mathbb RR(k)^p\,\frac{dk}{2\pi}. \tag{24}\] First truncate the exponentially decaying multiplier \(R\) to a bounded frequency interval. For any multiplier \(H\), \[\|PHP\|_1\le\operatorname{Tr}(P|H|P),\] by factoring through \(|H|^{1/2}P\) and using the Hilbert–Schmidt inequality. Equation (23) makes the normalized trace error tend to zero uniformly as the frequency interval grows; a telescoping product gives the same statement for each fixed power.

For a bounded compactly supported symbol \(S\), cyclicity of the trace writes the corresponding trace as \[\operatorname{Tr}\bigl(v^{-1}(Sv)P(SP)^{p-1}\bigr).\] The resolution \(\int_\mathbb R|k_x\rangle\langle k_x|\,dx=v^{-1}\) turns this trace into \(\int_\mathbb RF_N(x)\,dx\), where \[F_N(x)=\left\langle(Sv)P(SP)^{p-1}k_x,k_x\right\rangle_{\mathcal H}.\] The real-line evaluation vectors have a fixed norm. The multiplier \(Sv\) is bounded on the compact frequency support, and the rightmost operator acting on \(k_x\) contains \(P\). Hence \(|F_N(x)|\le C\|Pk_x\|\). Translate \(k_x\) to \(k_0\) and conjugate every \(P\) by the same real translation. The multipliers are unchanged. In the bulk \(|x|\le(1-\delta)L\), strong convergence of these translated projections therefore gives, uniformly, \[F_N(x)\mathrel{% \BeginAccSupp{method=hex,unicode,ActualText=27F6}% \longrightarrow% \EndAccSupp{}% } \left\langle S^pv k_0,k_0\right\rangle_{\mathcal H} =\int_\mathbb RS(k)^p\,\frac{dk}{2\pi}.\] The transition region has length \(O(\delta L)\); beyond \((1+\delta)L\), Equation (22) has an integrable exponential tail of size \(o(L)\). Divide by \(2L\), then let \(\delta\downarrow0\). This proves the compact-frequency limit and hence Equation (24).

On the real frequency axis, \(|R(k)|\le1\) and \(\Re R(k)\ge0\): here \(2\beta\Re\zeta=\pi/3\). Moreover \(\|PRP\|_1=O(L)\) by Equation (23). For \(0<t<1\), the logarithmic power series and Equation (24) yield \[\lim_{N\to\infty}\frac1{2L} \operatorname{Tr}\log(I+tPRP) =\int_\mathbb R\log(1+tR(k))\,\frac{dk}{2\pi}.\] Accretivity gives \(\|(I+tPRP)^{-1}\|\le1\) for \(t\ge0\), so the trace derivative is bounded by \(\|PRP\|_1=O(L)\). The same integrable bound holds for the limiting integral. Thus the limit extends to \(t=1\). Taking real parts is equivalent to taking the logarithm of the determinant modulus in Equation (20).

Finally shift the contour from \(\Re\zeta=\gamma/2\) to \(\Re\zeta=0\). The function \(\log(1+1/(2\cos(2\beta\zeta)))\) is analytic in the intervening strip, with the branch tending to zero at imaginary infinity; its horizontal-end contributions vanish exponentially. Therefore \[\int_\mathbb R\log(1+R(k))\,dk =\int_\mathbb R\log\left(1+\frac1{2\cosh(2\beta k)}\right)\,dk =\frac{5\pi^2}{72\beta}.\] For the last evaluation use evenness and set \(y=e^{-2\beta k}\) for \(k>0\). The logarithm becomes \(\log[(1-y^3)/((1-y)(1+y^2))]\). Termwise integration of its convergent logarithmic series, or monotone limits of their integrals, gives \[\frac1\beta\int_0^1 \frac{\log(1-y^3)-\log(1-y)-\log(1+y^2)}{y}\,dy =\frac1\beta\left(-\frac{\pi^2}{18} +\frac{\pi^2}{6}-\frac{\pi^2}{24}\right) =\frac{5\pi^2}{72\beta}.\] Combining this with \(2L/(2\pi)\) and \(\beta\gamma=\pi/3\) gives the coefficient \(5\pi/(72\beta\gamma)=5/24\) in Equation (17). ◻

Proof. On the main rows the identity follows from the displayed Fourier symbols. For an elementary far shift \(j\) in group \(\sigma\), divide the \(H\)-row versus near columns (own group first) by \(2\pi c_*/(k\sinh\pi k)\); it is \((\cosh(\beta k)e^{\sigma(4-j)\beta k},e^{\sigma(p-j\beta)k})\), as follows using \(2<j<5\). These entries give \(V\) after convolution for far shifts as well. Equality at nonzero frequencies suffices (matching real tails, with zero limiting difference, and bounded odd imaginary parts).

For a site pair, the sign-normalized log of \(B_s\) or \(B_c\) plus the corresponding entry of \(\varrho*H_{nn}*\varrho\) at real log difference equals the normalized log of \(H_*^2\) at the site log difference (with imaginary shift). For the same-group formula the symbol divided by \(2\pi/(k\sinh\pi k)\) on the left is \(-\cosh(2\beta k)-\cosh(\beta k)+\cosh(3\beta k)/c_*(k)\). On the right it is \(-2\cosh(\beta k)+\cosh(4\beta k)-\sinh(\pi k)\tanh(3\beta k/2)\), the same expression. To check the second expression, factor \(\cosh u+d\) and \(\sinh(u/2)\) in \(H_*\), and use the scaled \(K_\pi-K_0\) identity \[\widehat{\log|\tanh(\gamma u/2)|}(k) =-\frac{\pi}{k}\tanh\frac{\pi k}{2\gamma}.\] Cross entries acquire the factor \(e^{-\beta k}\) (extra sinh terms cancel since \(\sinh(3\beta k)/c_*=\sinh(2\beta k)+\sinh(\beta k)\)); \(H_*\) is analytic and nonzero through this shift, so its log acquires the same shift in the transform of the second derivative. Again real tails have matching slopes and zero constant, and imaginary parts are bounded odd. In particular the exponentiated half self term of \(\varrho*H_{nn}*\varrho\) is \[C_*=\frac{H_*(0)}{\sqrt{B_s(0)}}=\frac{(1+d)\gamma}{4d\cos(\beta/2)} .\] In \(B_s(0)\) the argument is the log difference. Thus all one-particle terms can be centered exactly using \(H\mu\); site pairs together with the mean factor \(\exp((\mu,H\mu)/2)\) yield \(\prod_{i<j}H_*(a_i-a_j)^2\) times \(C_*^N\) (up to fixed signs). ◻

Proof. Divide the real-kernel transform by \(\pi/(k\sinh\pi k)\); the matrix in group blocks has diagonals \(S\), off-diagonals \(T\). With \(h_j=\cosh(j\beta k/4)\), upper triangles listed by rows, they are \[\begin{split} S:\ &2(h_{16}-h_{12}+h_8),\ h_{13}-h_1+h_3+2h_5-h_9,\ h_{14}+h_2+2h_6;\\ &2(h_{16}-h_{12}+h_8),\ h_{13}+h_{15}+h_5+h_7;\\ &2(h_{12}+h_{16}+h_4+h_8).\\ T:\ &2(h_{12}+h_4-h_8),\ 2(h_1-h_3+h_7),\ 2(h_{10}+h_2);\\ &2h_{10},\ 2(h_{11}+h_9);\\ &4(h_{12}+h_8). \end{split}\] These follow by summing log-factor exponents with angles modulo \(2\pi\). Expand in \(v=\cosh(\beta k/4)-1\). At zero \(S=T=2w w^{\mathsf t}\), \(w=(1,1,2)^{\mathsf t}\). All coefficient matrices of \(S\pm T\) of positive degrees through 16 are strictly positive definite after division by the corresponding coefficient of \(h_{16}\). Use the ratios \[R_i(l)=i^2\prod_{h=1}^{l-1}(i^2-h^2)\big/\big(16^2\prod_{h=1}^{l-1}(16^2-h^2)\big)\] by the cosh polynomial differential equation \((2v+v^2)h_i''+(1+v)h_i'=i^2 h_i\) as polynomials in \(v\). For clarity the first three normalized \(S,T\) triangles are: \[\begin{array}{c|lll} l&S_{11},S_{12},S_{13}&S_{22},S_{23}&S_{33}\\\hline 1 &11/8,73/128,17/16&11/8,117/64&15/4\\ 2 &203/136,483/1360,849/1360&203/136,1703/1360&939/340\\ 3 &685/408,5849/22848,23/51&685/408,3163/3264&1133/476 \end{array}\] \[\begin{array}{c|lll} l&T_{11},T_{12},T_{13}&T_{22},T_{23}&T_{33}\\\hline 1 &3/4,41/128,13/16&25/32,101/64&13/4\\ 2 &35/68,19/272,413/1360&165/544,175/272&513/340\\ 3 &9/28,293/22848,55/476&55/476,2035/7616&155/204 \end{array}\] Both sum and difference have positive successive leading minors (they exceed \(1/2,1/4,1/10\) respectively by substitution). For degrees \(l\ge4\), using \(0\le R_i(l)\le R_i(4)\) gives absolute off-diagonal bounds \(0.21,0.39,0.90\) and diagonal lower bounds \(1.60,1.75,1.59\), enough by diagonal dominance. This coefficient calculation gives the asserted norm comparison as follows. Near zero, the constant rank-one term contributes \(\|D\|_2^2\), and the strictly positive degree-one coefficients control the remaining directions at constant scale. On each compact frequency interval avoiding zero, positivity and continuity give uniform upper and lower bounds. At large \(|k|\), the \(h_{16}=\cosh(4\beta k)\) term is diagonal and dominates all other entries. Since \(\pi=4\beta\), multiplication by \(\pi/(k\sinh\pi k)\) gives the scale \(|k|^{-1}\) there. These three ranges prove the two-sided Sobolev bound. Finally \(\widehat d_x=ik\widehat D\), and \(d_x=\sum_Iw_Ix_I\), so the same bound controls the stated \(H^{1/2}\)-norm of \(D\). ◻

Proof. Restoring the omitted self terms of \(H_{nn}-H_{\rm ref}\) contributes per interior variable \[(c/d)\,2\sin t\,t'(u),\] since \(E_s(u)\sim(d/c)^2u^2\). Combined with \(du/(2\pi)\), this is \(c/d\) times the one-particle reference measure. The scalar cancellation is \[(2d)C_*(c/d)=\gamma/2.\] Replacing \(\widetilde H\) by \(H\) gives exactly the additional factors \(A\) and \(\exp((\mu,R_h\zeta))\). The latter is at most \(\exp(C_hn_e)\): the cutoff can meet \(\mu\) only within distance one of an exterior main particle, where \(|u|\ge L-1\), and its logarithmic singularity is integrable. Missing \(c/d\) factors and far activities each cost a fixed constant per explicit variable. Finally the sector sign calculation leaves precisely the phase \(e^{i(\eta+\pi f)M}\), with all remaining signs independent of \(M\) at fixed \(\zeta\). This proves both the identity and the prefactor bound. ◻

Proof. At nonzero frequency, \[|\widehat{\widetilde H}_{ne}-\widehat H_{ne}\widehat\psi| \le C h^{c_0}(1+|k|)^{-1}\widehat\psi(k)\] for some \(c_0>0\), with no residual distributions at zero. Indeed nonsingular terms have transform \(O(e^{-c|k|}(1+k^{-2}))\), multiplied by \(1-\widehat\psi\). For the singular \(S_0\) term the difference in real space is \((1-\chi)(S_0-S_0*\psi)\) with derivatives in \(L^1\) bounded by \(O(h^2)C^{l+1}(l!)^2\) by even averaging of imaginary shifts off the singularity, and the cutoff bounds. This gives \(O(h^2)\exp(-c\sqrt{|k|})\) in frequency, sufficient by the lower bound for \(\widehat\psi\). Thus the mixed-kernel error is bounded by \(C h^{c_0}\|\alpha-\mu\|_{-1/2}\|\psi*\zeta\|_{-1/2}\). For explicit-explicit pairs only the real kernel matters. Every smooth real summand (apart from diagonal \(S_0\)) changes under \(\psi*\psi\) averaging by \(O(h)e^{-c|u|}\), using exponentially decaying second derivatives and the tail and second moment of \(\psi\). The diagonal singularity can only decrease. And the missing same-variable diagonals in the smoothed model cost \(O_h(n_e)\).

To absorb these errors we need a quadratic count bound that is valid even when the interior trial densities are signed. We obtain it by testing far labels separately and testing main labels only outside the wall, where those trial densities vanish. Partition the explicit coordinates into unit-scale cells and let \(n_j\) denote their variable counts. We claim \[ \sum n_j^2\le C(1+|\log h|)(B(x)+1). \tag{26}\] For far labels use translates of a nonnegative fixed bump covering a unit interval, to test separately each \(x_I=\psi*\zeta_I\) positive. For outer main counts on either side of the wall use bumps supported outside the wall, adapted to cells starting from the wall (the grid may change by bounded overlaps). For the first cell use a bump rising from zero at the wall to 1 within distance \(h\), plateauing to distance 2, then cutting off. This captures at least a constant smeared mass from each unsmeared point of the first cell, by the displayed real kernels. On all these outer tests \(\operatorname{Re}\alpha\) does not intervene and \(\mu\) has cell masses summably bounded. The \(H^{1/2}\) norms squared of the bumps are \(O(1+|\log h|)\) even at the wall, by the kernel seminorm \(\int|b(u)-b(v)|^2|u-v|^{-2}du\,dv\) (equivalence by Plancherel); for the transition of length \(h\), the translation differences in \(L^2\) squared cost \(O(\min(l,l^2/h))\) for translations by \(0<l<1\). Bounded overlap gives the same squared-norm bound times \(\sum a_j^2\) for linear combinations (sum differences pointwise has bounded multiplicity). Duality proves (26).

Also \(\|\psi*\zeta\|_{-1/2}^2\le C(1+|\log h|)\sum n_j^2\). Indeed the spatial kernel for multiplier \((1+k^2)^{-1/2}\) is bounded by \(C(1+|\log|u||)e^{-c|u|}\), by writing the symbol as a gamma-integral of Gaussians. Convolved twice with \(\psi\) it is bounded by \(C(1+|\log h|)e^{-c'|u|}\) using the density and tail bounds. This and (26) control the mixed error just found (substitute \(\alpha-\mu=x+i z-\psi*\zeta\)), and the \(O(h)e^{-c|u|}\) explicit pair errors, proving (25). ◻

Proof. Outside the wall \(\operatorname{Re}\alpha\) is absent. For \(t>L\), the total count beyond \(t\) is at most twice its positive smeared cumulative there, hence its square integrates on \([L,\infty)\) to \(O(B(x)+1)\) by \(\|D\|^2\) and the \(\mu\)-tail. Same on the other side. The cumulative counts are nonnegative integers, so their squares dominate the counts themselves. Integrating gives the distance sum bound. It also bounds exterior counts away from the wall; (26) controls the remaining cells near the wall and the far-particle cells inside it.

Total smeared mass outside the wall is at most \(C_h(1+\sqrt{B(x)})\) also, by the cumulative test outside distance one from the wall and (26) convolved by \(\psi\) for the first cell. Finally \(\int r=-\operatorname{Re}M\); test on the wall region with a smooth plateau extending outside by distance one, cost \(O(\sqrt L)\) in \(H^{1/2}\), and bound the remainder by the positive tail masses. In particular, retaining a fraction of \(B(x)\) in a trial lower bound gives exponential damping \(\exp(-c_h\ell_e)\), sufficient for convergence and tail truncation of the explicit absolute integrals. Summing their factorial measures with factors \(\exp(C_hn_e)\) costs at most \(\exp(O_h(L))\): far variables have effective volume \(O(L)\), and exterior main variables have effective volume \(O(1)\). This is the coarse cost; part 4 of Proposition 18 will improve it for the configurations needed by the exponent. ◻

Proof. For part 1, set \(z=0\) in (28). The remaining linear term satisfies \[|b_1\textstyle\int r|\le C_h\sqrt{L(B(x)+1)} \le \tfrac14B(x)+C_hL\] by Corollary 17; choose \(h\) so that the smoothing loss consumes at most another fixed fraction of \(B(x)\). This proves the first bound without any count truncation.

For part 2 suppose that \(n_e\) is polynomially bounded and use the stated norm and truncation properties of \(p_L\). For \(z_\sigma=-\sigma v p_L,\ v=\operatorname{Im}M\), (28) gives \[ \mathcal A\ge c B(x)-C_h L-C_h n_e+2\pi jv-C_h(v^2+(1+f)|v|). \tag{29}\] Here \(D_z=0\), and the mixed baseline and odd-remainder errors cost \(C|v|\sqrt{B(x)}\). This bound uses no polynomial restriction on \(j\).

For part 3 suppose also \(|j|\le{\rm poly}(L)\), with the polynomial fixed. We select \(v=\operatorname{Im}M\), independent of \(\operatorname{Re}M\), together with \(z\). In the choices below we first specify smooth \(z_r,D_z\) supported on \(|u|\le(1-\delta)L\) (and \(v=-\int z_r\)); then we actually add \(v(p_L^\circ-p_L)\) to \(z_r\), leaving \(D_z\) unchanged. Thus each difference \(z_\sigma+\sigma v p_L\) is a smooth density derivative with primitive compactly supported in \(|u|\le(1-\delta/2)L\); all its required sizes and fixed-order derivatives are polynomially bounded. In evaluating (28) we can neglect the exponentially small correction, at cost \(o(1)(1+\sqrt{B(x)})\) by the norm bounds, total variation, and the polynomial count bounds.

Take smooth nonnegative probes \(\varphi_\pm(u)\le1\), supported on \(\pm[(1-2\delta)L,(1-\delta)L]\), rescaled fixed bumps supported away from their interval edges with mass comparable to \(\delta L\). Set \(\theta_\pm\) equal to the corresponding weighted averages of \(\Theta\); let \(U=\max|\theta_\pm|\). A first, coarse tilt uses \[D_z=0,\qquad z_r=-t_0\sum_\pm\theta_\pm\varphi_\pm\] with small fixed \(t_0>0\). Then \(-\int\Theta z_r=t_0\sum\theta_\pm^2\int\varphi_\pm\). The \(B(z)\) cost is at most \(C t_0^2\sum\theta_\pm^2\int\varphi_\pm\); baseline and odd mixed terms are absorbed similarly with a fraction of \(B(x)\). The term with \(b_1\) is absorbed as in the real bound. Thus \[ \mathcal A\ge cB(x)+c_\delta U^2 L-C_h L-C_h n_e . \tag{30}\] By positivity, the number of far particles on \([-(1-2\delta)L,(1-2\delta)L]\) is \(\le C U\) for large \(L\): their smeared masses contribute at least a constant each to the drop between the two probe averages.

For part 4 suppose \(U\le U_0\), where \(U_0\) may be arbitrarily large but fixed. Given a desired small error, choose a small parameter \(\varepsilon>0\), then \(h\) sufficiently small (so \(s(h)\ll\varepsilon\)), then \(\delta>0\) sufficiently small relative to these. \(U_0\) can be chosen after these parameters. Use a smooth mask \(0\le w(u)\le1\) supported on \([-(1-3\delta)L,(1-3\delta)L]\), vanishing at distances \(\le\sqrt L\) of far positions, equal to 1 except in boundary buffers of total width \(O(\delta L)\) inside \([-L,L]\) and holes of total width \(O_{U_0}(\sqrt L)\). Take fixed-order positive derivatives \(O_{U_0,\delta,l}(L^{-l/2})\). This is possible by the count conclusion. Put \(s(u)=\operatorname{sign}(a+2\pi(j+f_R^0(u)))\) and \(A_0=|a|/(2\kappa),\ B_0=-p b_1/(2\kappa)\). Use \[z_r=-A_0\,s(u)w(u),\qquad D_z=B_0 w(u),\] with the correction convention above. These are smooth before correction even at jumps. Since \(|a|<\pi\), \(|a+2\pi(j+f_R^0)|\ge |a|\). On the mask support all far smearing tails from points outside their own respective side of \(u\) are superpolynomially small. Consequently \(-\int\Theta z_r\ge |a| A_0\int w-o(1)\), and \(\sum g_I\int x_I D_z=o(1)\). Moreover \[B(z)\le (B_0^2+(\kappa-p^2)A_0^2)\int w+o(L),\qquad |(x,J_o z)|\le o(\sqrt L)\sqrt{B(x)} .\] Indeed before correction the scaled \(L^2\) masses of \(z_r,D_z\) (squared divided by \(L\)) concentrate at low frequency since their positive-order derivatives have norms \(o(\sqrt L)\). Use the symbol expansion and the upper norm bound, and the odd-remainder multiplier bound.

We give details for the real-field optimization so that no slow-variation assumption on \(x\) is needed. Let subscript \(\ell\) mean a smooth compact low-pass Fourier cutoff equal to 1 near zero, even real with values in \([0,1]\), supported on sufficiently small fixed frequencies depending on \(\varepsilon\). By the positive small-frequency expansion, reserving a fraction on all frequencies, \[B(x)\ \ge\ c_\varepsilon B(x) +(1-C\varepsilon)\int \big(D_\ell^2+Q(r_\ell,x_{{\rm far},\ell})\big).\] On the mask \(x_{{\rm far},\ell}\) is superpolynomially small (Schwartz kernel tails, count bound). Thus in this lower bound we can use \((1-C'\varepsilon)\int w(D_\ell^2+(\kappa-p^2)r_\ell^2)-o(1)\) besides the reserve. The pairings against \(ws,w\) can replace \(D,r\) by \(D_\ell,r_\ell\) at \(o(\sqrt L)\sqrt{B(x)}\) cost by the derivative bounds and Sobolev norms. Also \[|\int(1-w)r|\ \le\ (C\sqrt{\delta L}+o(\sqrt L))\sqrt{B(x)}+C_h .\] Use the smooth wall plateau of (27) minus \(w\), whose \(L^2\) norm has the stated bound by the support estimate and whose derivative norm is \(o(\sqrt L)\), then positive outer tail bounds for the rest. Constants depending on \(U_0,\delta,h\) in these estimates are allowed inside little-oh terms.

Absorb errors with a portion of the reserve, including (25), losing \(C\varepsilon L+o(L)\) with \(\delta\) sufficiently small. It remains on measure \(w\,du\) to complete squares for \(D_\ell,r_\ell\), whose linear coefficients in (28) are \(2p A_0 s,\ -(b_1+2p B_0)\). The lower cost from these optimizations per unit mass is at worst \[-p^2 A_0^2-\frac{(b_1+2p B_0)^2}{4(\kappa-p^2)}-O(\varepsilon).\] Adding \(-B_0^2-(\kappa-p^2)A_0^2+|a| A_0\) leaves \((a^2-b_1^2)/(4\kappa)\) before errors. Thus \[ \mathcal A\ \ge\ c'_\varepsilon B(x)+2L\,\frac{\operatorname{Re}(\eta^2)}{4\kappa} -C\varepsilon L-o(L)-C_h n_e. \tag{31}\] The retained constant can be taken independent of \(\delta,U_0\). For these bounded-\(U\) estimates the explicit factorial measures with \(\exp(C_h n_e)\) and a portion of the retained damping cost at most \(\exp(C_{h,\varepsilon}\delta L+o(L))\): outer main points cost bounded grand mass by tail damping, only a bounded number of far points can be on the interior interval just counted, and the remaining far points use an effective volume \(O(\delta L)+O_{h,\varepsilon}(1)\) per variable, with factorial divisors. This also works with more fixed exponential factors per particle, by further reducing \(\delta\). Bounds (29)–(31) are deterministic bounds on trial densities. Implementing them for the complex integrals, including fluctuation determinants and the discreteness of \(M\), takes further work below. ◻

Proof. First suppose \(|s-v|>c_0>0\). The sinh terms have the required estimates directly. Split the cos-difference logarithm into its two sine logarithms. Then \(\sin(|\theta(s)-\theta(v)|/2)\) is bounded below by \(c\max_{[s,v]}\lambda_L\), reversing the interval if needed, and \(\sin((\theta(s)+\theta(v))/2)\) is at least as large. One derivative in each variable supplies the decaying ratio \[\frac{\lambda_L(s)\lambda_L(v)}{(\max_{[s,v]}\lambda_L)^2}.\] Higher derivatives preserve this bound, proving the off-diagonal estimate for \(\mathscr K\).

Near the diagonal and away from the endpoints, divide both \(2\sinh((W(s)-W(v))/2)\) and \(\cos\theta(s)-\cos\theta(v)\) by \(s-v\). Their natural local scales are respectively \(1\) and \(\lambda_L^2\), since \(\sin\theta\asymp\lambda_L\) there. Averaging derivatives along the joining interval gives uniform Gevrey bounds and lower bounds on those scales. Their logarithmic ratio is therefore smooth.

Near a common endpoint, let \(d_s,d_v\) be the distances from that endpoint. Both differences have the additional factor \(d_s+d_v\): even reflection makes their zeros occur at both \(s=v\) and the reflected diagonal. Divide by \[(s-v)(d_s+d_v).\] The division is smooth Gevrey, obtained by averaging a derivative of the first divided difference from its reflected zero. The quadratic fold \(\omega'(d)\asymp d\) and the sine factors give positive lower bounds for both quotients on their respective scales. The same log-ratio argument now proves regularity through the corner. Even reflection in either variable follows from the construction of \(W\) and \(\theta\).

Finally, in variables at distance at least \(\delta L\) from the endpoints, the compensated kernels continue analytically for uniformly small complex displacements, with the same fixed-order derivative control in the remaining variables. Here \(W(s)=s\) and \(\theta'=C_L\pi\varrho\) is analytic with relative bounds on uniformly small disks, so the divided-difference arguments apply there as well. ◻

Proof. For each main component, decompose the centered mean as \[m'_\sigma p_L-m\varrho =(m'_\sigma C_L-m)\varrho +m'_\sigma(p_L-C_L\varrho).\] The coefficient in the first term is polynomially bounded, since \(m'_\sigma-m\) is polynomial and \(C_L-1=O(e^{-\gamma L})\). Its convolution is controlled by \(H_{nn}*\varrho=V\).

The two measures in the second term agree in the core. Near or beyond the ends their densities, written in \(ds\) for \(p_L\) and \(du\) for \(\varrho\), have derivative bounds with a factor \(O(e^{-\gamma L})\) and exponentially bounded tails. This factor cancels \(m'_\sigma=O(e^{\gamma L})\). Insert a fixed-width physical Gevrey cutoff there. For convolution of the first density with the diagonal logarithm, subtract \(-2\log|s-v|\) and its reflected-image term using the divided differences of \(W\). The remaining kernel is smooth. Transfer derivatives of the ordinary translated logarithm to the compactly cut off, evenly reflected density. For convolution with the second density, transfer derivatives in physical \(u\) and then compose with \(W\). Both terms are analytic in the deep core by separation from their supports.

The source \(\widetilde H_{ne}\zeta\) has the same bounds for fixed \(h\). Its regularization concerns only exterior main interactions and leaves an analytic kernel in the core. These arguments also give polynomial total variation and moments for \(m'p_L-\mu\), hence the stated constant-term bound by log integrability. Finally, real logarithmic singularities of explicit pairs are repulsive; away from them the kernels have at most linear growth. The polynomial count and excursion bounds therefore give \(\operatorname{Re}E_e\ge-\operatorname{poly}(L)\), including when distinct explicit particles approach each other. ◻

Proof. Partition \([-A,A]\) into intervals \(I_a\) of lengths in \([1,2]\), and use normalized cosine bases \(e_{a,n}\), \(n\ge0\), on these intervals. The mixed-kernel estimates imply \[\left|\big\langle e_{a,n},\mathscr K e_{b,m}\big\rangle\right| \le \frac{Ce^{-c|a-b|}}{(1+n^2)(1+m^2)}.\] For a positive frequency, integrate twice by parts: the first boundary term vanishes because the sine is zero at either endpoint, and the second boundary term involves only the first derivative of the kernel. Doing this in both variables gives the displayed estimate, including all derivative boundary terms. Summing the trace norms of these rank-one matrix entries gives \(\|\mathscr K\|_1\le CL\). Schur’s test applied directly to its exponentially decaying kernel gives \(\|\mathscr K\|\le C\). The bounds for \(T\) follow from the uniform bound for \(j\).

It remains to control the lower edge of the spectrum. For a two-component field, sum over components in the following definition: \[\mathfrak e(Q)= \iint_{|s-v|\le2}\frac{|Q(s)-Q(v)|^2}{|s-v|^2}\,ds\,dv +\int_{d(s)\le2}\frac{|Q(s)|^2}{d(s)}\,ds, \qquad d(s)=A-|s|.\] We first establish the two comparisons \[h_0(Q,Q)\le C\bigl(\mathfrak e(Q)+\|Q\|_{L^2(ds)}^2\bigr), \qquad \langle Q,A_TQ\rangle_{\mathcal X}\ge c\mathfrak e(Q).\] The circle difference formula for the odd extension in \(t\) expresses \(h_0\), up to fixed constants, using the two kernels \[\frac{\theta'(s)\theta'(v)} {\sin^2((\theta(s)-\theta(v))/2)},\qquad \frac{\theta'(s)\theta'(v)} {\sin^2((\theta(s)+\theta(v))/2)},\] paired respectively with \(|Q(s)-Q(v)|^2\) and \(|Q(s)+Q(v)|^2\). When \(|s-v|\ge c_0\), the first kernel is exponentially integrable in the separation. The second kernel has the same property except at a common endpoint; there it is bounded by a constant times the inverse square of the sum of the endpoint distances. Its integral in one variable is at most \(C/d(s)\). These assertions follow from \(\lambda_L(s)\asymp e^{-\gamma|s|}\) and \(\sin\theta(s)\asymp\lambda_L(s)\min(1,d(s))\). On the remaining diagonal region the first kernel is bounded by \(C|s-v|^{-2}\). This proves the upper comparison.

For the lower comparison, push the primitive forward to physical coordinate \(u=W(s)\) and extend it by zero outside \([-L,L]\). Lemma 14 bounds its Fourier seminorm with multiplier \(k^2/(1+|k|)\). For any fixed sufficiently large \(C_0\), this multiplier is comparable to the multiplier \[\int_{|z|\le C_0}\frac{|e^{ikz}-1|^2}{z^2}\,dz \asymp\frac{k^2}{1+|k|}\] of the truncated difference seminorm \[\iint_{|u-u'|\le C_0} \frac{|Q(W^{-1}(u))-Q(W^{-1}(u'))|^2}{|u-u'|^2}\,du\,du',\] where the zero extension is understood. For an interior point at distance \(\ell<C_0/2\) from a wall, integration against the adjacent exterior interval gives weight \(1/\ell-1/C_0\asymp1/\ell\). Thus interactions with the exterior zeros give a wall weight comparable to \(W'(s)/(L-|W(s)|)\asymp1/d(s)\) near an endpoint. When the two endpoint distances are comparable, \(W'(s)W'(v)/|W(s)-W(v)|^2\asymp|s-v|^{-2}\). The part with noncomparable distances is bounded by the wall term: for example, integrate \(|s-v|^{-2}\) over \(d(v)<d(s)/2\) or \(d(v)>2d(s)\), using \(|Q(s)-Q(v)|^2\le2|Q(s)|^2+2|Q(v)|^2\). A fixed enlargement of the endpoint region causes no change in these bounds. This proves the second comparison.

For completeness, divide any interval of length comparable to \(b\ge1\) into cells of lengths in \([1/2,1]\), and let \(q_a\) be the mean of \(Q\) on cell \(a\). The local seminorm controls \[\sum_a\int_{I_a}|Q-q_a|^2 +\sum_a|q_{a+1}-q_a|^2.\] The first bound is the elementary variance identity on a cell; the second follows by averaging the difference over two adjacent cells, whose mutual distances are at most \(2\). If the mean on the whole block vanishes, the discrete Poincare inequality gives \(\sum_a|q_a|^2\le Cb^2\sum_a|q_{a+1}-q_a|^2\). Thus \(\|Q\|_2^2\le Cb^2\mathfrak e(Q)\) for fields with zero mean on each such block. On the whole interval, the wall term controls the first and last cell means; summing the successive differences then gives the same estimate with \(b\asymp L\), without any constraints. The two comparisons now give the gap \(cL^{-2}\).

For \(t\) above a fixed multiple of \(L^{-2}\), choose \(b=c_1t^{-1/2}\) with \(c_1\) sufficiently small. Vanishing block means impose at most \(CL/b\) linear constraints, and on their common kernel the Rayleigh quotient of \(A_T\) is at least \(t\). The min–max principle proves the counting bound. Below the global gap the count is zero; the intervening range is handled by changing the constants. Compressing a positive Hermitian form cannot increase this count. Finally, if a subspace satisfies \(\|TQ\|<t\|Q\|\), then on it \(\langle Q,A_TQ\rangle\le\|Q\|\|TQ\|<t\|Q\|^2\). The variational characterization of singular values therefore gives the same count and gap for \(T\) and its compressions. ◻

Proof. Accretivity gives an injective operator with closed range. Since \(T-I\) is trace class, \(T\) has Fredholm index zero and is therefore invertible. Its Fredholm determinant is defined by the convergent exterior-power series. Equivalently, take determinants of finite orthogonal compressions converging strongly to the identity, then pass to the limit in trace norm. The same procedure gives \(\log|\det T|=\tfrac12\operatorname{Tr}\log(T^*T)\). We compute this trace through the limiting symbol \[T_\infty(k)= \frac{k^2\widehat H_{nn}(k)} {2\pi k\coth(\pi k/(2\gamma))}.\]

First consider a nonconstant word in \(T-I\) and \(T^*-I\). By cyclicity its trace is an integral alternating kernels \(G\) and \(\mathscr K\) or \(\mathscr K^*\). The local bound \(C(1+|\log|s-v||)\) for \(G\), and exponential off-diagonal bounds for all kernels, provide an integrable majorant for the relative coordinates in this cyclic integral. More explicitly, truncate every separation at a fixed radius \(R\). The discarded integral is bounded by \(CL e^{-cR}\) times a fixed polynomial in \(R\). Each local logarithmic singularity can be integrated first against an adjacent bounded \(\mathscr K\) kernel. With \(R\) fixed, the central region and the two endpoint regions of width \(O(R)\) contribute \(O(R)\), which vanishes after division by \(2L\).

Under translation into either remaining tail, with distance from both the center and the endpoints tending to infinity, one has \(W(s)=s\) and \(\theta(s)\) or \(\pi-\theta(s)\) asymptotic to a constant times \(e^{-\gamma|s|}\), locally with every fixed derivative. Consequently \[G(s,v)\mathrel{% \BeginAccSupp{method=hex,unicode,ActualText=27F6}% \longrightarrow% \EndAccSupp{}% } \frac{1}{2\pi^2}\log\coth\frac{\gamma|s-v|}{2}.\] The mixed derivative of the reference kernel tends to that of \(-2\log|2\sinh(\gamma(s-v))|\); its additional one-variable terms disappear upon taking mixed derivatives. The Fourier transforms of these two limiting kernels are reciprocal, respectively \[\frac{\tanh(\pi k/(2\gamma))}{2\pi k}, \qquad 2\pi k\coth\frac{\pi k}{2\gamma}.\] It follows from the logarithmic transform identities that multiplication by the limiting \(\mathscr K\) symbol gives \(T_\infty-I\). Dominated convergence, followed by \(R\to\infty\), proves that the trace of each word, divided by \(2L\), converges to the integral of the corresponding matrix trace against \(dk/(2\pi)\).

We pass from words to the logarithm without losing uniformity at zero. For singular values \(s_i(T)\), Equation (32) gives \[\sum_{s_i<\tau}\log\frac{\tau}{s_i} =\int_0^\tau\frac{\#\{s_i<t\}}{t}\,dt \le CL\sqrt\tau.\] Thus clipping \(\log(T^*T)\) below \(\tau^2\) changes its normalized trace by at most \(C\sqrt\tau\). On the compact interval containing its spectrum, approximate the clipped logarithm by polynomials vanishing at one, with error at most \(\epsilon|y-1|\): apply polynomial approximation to the continuous quotient by \(y-1\). The resulting trace error is bounded by \(\epsilon\|T^*T-I\|_1\le C\epsilon L\). The symbol has the same integrable control: its smaller singular value is comparable to \(k^2\) near zero, its other singular value stays positive there, and it tends exponentially to the identity at infinity. These assertions follow directly from the displayed \(2\times2\) symbol and its positive Hermitian part. Hence clipping costs \(O(\sqrt\tau)\) on the symbol side as well. Passing successively \(L\to\infty\), \(\epsilon\downarrow0\), and \(\tau\downarrow0\) yields \[\lim_{L\to\infty}\frac{\log|\det T|}{2L} =\int_{\mathbb R}\log|\det T_\infty(k)|\,\frac{dk}{2\pi}.\]

Set \(x=\beta|k|\) and \(y=e^{-x}\). The determinant is positive, and its positive square root equals \[\sqrt{\det T_\infty(k)} =\frac{c_*(k)\sinh(3x)}{\sinh(4x)\coth(3x/2)} =\frac{(1-y^3)(1-y^6)}{(1+y)(1-y^8)}.\] Integrating the absolutely integrable logarithmic series gives \[\begin{split} \int_0^\infty\log\sqrt{\det T_\infty(x/\beta)}\,dx &=-\frac{\pi^2}{18}-\frac{\pi^2}{36} -\frac{\pi^2}{12}+\frac{\pi^2}{48}\\ &=-\frac{7\pi^2}{48}. \end{split}\] Here \(\int_0^\infty\log(1-e^{-ax})\,dx=-\pi^2/(6a)\) and \(\int_0^\infty\log(1+e^{-x})\,dx=\pi^2/12\), obtained by termwise integration; \(\sum n^{-2}=\pi^2/6\) follows, for example, from Parseval’s identity. Accounting for both signs of \(k\), the square of the displayed square root, and \(\gamma=\pi/(3\beta)\), we obtain \[\log|\det T|=-\frac{7\pi}{12\beta}L+o(L) =-\frac{7\gamma}{4}L+o(L),\] which is Equation (33). ◻

Proof. We first approximate \(\mathscr K\) on \(L^2(ds)\). Truncate the cosine basis in each unit-scale cell at frequency \(J\). The coefficient estimate in Lemma 21 gives trace-norm error at most \(CL/J\), since \(\sum_{n>J}(1+n^2)^{-1}\le C/J\) and the sum over cell separations converges. Approximate the finitely many retained cosines in each cell by smooth functions compactly supported inside that cell, with \(L^2\)-error at most \(\eta\), and orthonormalize. For fixed \(J\) and sufficiently small \(\eta\), their Gram matrices are uniformly invertible. Replacing a rank-one term \(u\otimes v\) by its smoothed counterpart changes its trace norm by at most \(\|u-\widetilde u\|\|v\|+ \|\widetilde u\|\|v-\widetilde v\|\). Summing the absolutely convergent cell expansions therefore adds at most \(C_J\eta L\).

Discard cells meeting \(|s|>(1-2\delta_*)L\). There are \(O(\delta_*L+1)\) such cells, and the total trace-norm contribution of terms with either index in these cells is of that same order. Denote the resulting real bump projection by \(Q_b\). Then \[\|\mathscr K-Q_b\mathscr KQ_b\|_1 \le C\bigl(J^{-1}+C_J\eta+\delta_*\bigr)L+O(1).\] Choose \(J\), then \(\eta\), then \(\delta_*\), so the coefficient on the right is as small as required. There are \(O_\xi(L)\) bumps; each has uniformly bounded derivatives of every fixed order, with constants allowed to depend on these fixed choices.

Let \(b_1,\ldots,b_d\) be the orthonormal bump basis and put \(v_i=j^*b_i\). As a function in \(s\), \(v_i=Gb_i\). Their Gram matrix in \(\mathcal X\) is uniformly conditioned. The upper bound follows from \(\|j\|\le C\). For the lower bound, every \(b\) in the bump span is a permissible smooth field and satisfies \(\|b\|_{\mathcal X}\le C_\xi\|b\|_2\), by the local upper norm estimate and its uniformly bounded \(H^1/L^2\) ratio. Duality gives \[\|j^*b\|_{\mathcal X} \ge\frac{|\langle j^*b,b\rangle_{\mathcal X}|} {\|b\|_{\mathcal X}} =\frac{\|b\|_2^2}{\|b\|_{\mathcal X}} \ge c_\xi\|b\|_2.\] Thus the map \(b\mathrel{% \BeginAccSupp{method=hex,unicode,ActualText=21A6}% \mapsto% \EndAccSupp{}% }Gb\) loses no linear independence or uniform conditioning.

Choose a smooth cutoff equal to one on a neighborhood of the bump support and zero outside \(|s|\le(1-\delta_*)L\), leaving a separation at least \(c\delta_*L\) between the bumps and its nonconstant region. Multiply each \(v_i\) by this cutoff, obtaining \(\widetilde v_i\). The off-diagonal derivative bounds for \(G\) imply \[\|v_i-\widetilde v_i\|_{\mathcal X} \le C_\xi L^C e^{-c\delta_*L}.\] To check the endpoint term, the explicit sine ratio gives \[G(s,v)=O\bigl(d(s)e^{-c|s-v|}\bigr)\] when \(s\) is close to an endpoint and \(v\) lies in a retained cell. Every fixed positive-order derivative has the same exponential bound, without necessarily retaining the factor \(d(s)\). The difference field vanishes at the endpoint. In particular, the elementary estimate \[\int_0^2\frac{|q(d)|^2}{d}\,dd \le 2\int_0^2|q'(d)|^2\,dd,\qquad q(0)=0,\] controls its wall contribution. The upper local norm estimate, or its consequence \(h_0(q,q)\le C\|q\|_{H^1(ds)}^2\) for such fields, then proves the claim. The Gram matrix remains uniformly invertible because \(d=O_\xi(L)\). If \(P_0,P\) are the orthogonal projections onto the spans of \(v_i,\widetilde v_i\), respectively, the formula \(V(V^*V)^{-1}V^*\) for a span projection yields \[\|P-P_0\|\le C_\xi L^C e^{-c\delta_*L}.\] The operator \(B=j^*Q_b\mathscr KQ_bj\) has both its range and its adjoint range in \(\operatorname{ran}P_0\); hence \(B=P_0BP_0\). Writing \(A_1=T-I\), we obtain \[\|A_1-PA_1P\|_1 \le2\|A_1-B\|_1+2\|P-P_0\|\,\|B\|_1 \le C\bigl(J^{-1}+C_J\eta+\delta_*\bigr)L+o(1).\] This proves (34) after the parameter choices above.

Finally, \(Gb_i\) has controlled derivatives of any fixed order in the core. Locally subtract the ordinary diagonal logarithm from \(G\); the remaining divided differences are smooth with uniform derivative bounds. For the logarithm itself, transfer derivatives to the smooth bump, and use the exponentially decaying off-diagonal estimates on the rest. Multiplication by the cutoff preserves these bounds. Passing to an orthonormal basis uses the inverse square root of a uniformly conditioned \(d\)-dimensional Gram matrix; its operator norm is bounded, so pointwise derivative bounds increase by at most \(O(\sqrt d)\). For the action of \(h_0\), write the test density as the derivative of \(H_{\rm ref}\) applied to the derivative of the core field. Subtract its local logarithm and transfer derivatives to that smooth input. The even endpoint reflections of the kernel give odd reflections of the resulting test density. Adding the smooth operator \(\mathscr K\) gives the same assertions for \(H_{\rm form}\). ◻

Proof. The gap for \(T_{SS}\) makes it invertible. Set \[Jy=(-T_{SS}^{-1}T_{SR}y,y),\qquad y\in R.\] Then \(TJy=(0,T_ey)\), so \[J^*(\operatorname{Re}T)J=\operatorname{Re}T_e, \qquad \|Jy\|\ge\|y\|.\] The gap follows immediately. If \(E\subset R\) is a subspace on which the Rayleigh quotient of \(\operatorname{Re}T_e\) is less than \(t\), the quotient of \(\operatorname{Re}T\) on \(JE\) is also less than \(t\). Since \(J\) is injective, min–max proves the count. These identities use Hermitian adjoints; the matrices of the bilinear energy are, separately, complex symmetric in a real basis.

We make explicit how the counting bound controls logarithms. If the positive eigenvalues or singular values of any operator under consideration have counting function \(N(t)\le CL\sqrt t\), then \[\sum_{s_i<\tau}\log\frac{\tau}{s_i} =\int_0^\tau\frac{N(t)}t\,dt\le2CL\sqrt\tau.\] The lower gap justifies the identity even for the finite-size operators. Put \(B=T_{SS}\oplus I_R\). Equation (34) gives \(\|T-B\|_1\le\xi L\), and \(\|T\|,\|B\|\le C\), whence \(\|T^*T-B^*B\|_1\le C\xi L\). The function \(y\mathrel{% \BeginAccSupp{method=hex,unicode,ActualText=21A6}% \mapsto% \EndAccSupp{}% }\log\max(y,\tau^2)\) has Lipschitz constant \(\tau^{-2}\). The trace difference of this function on two positive operators is bounded by that constant times their trace-norm difference; this follows first for finite matrices by the eigenvalue variation inequality and then by finite compressions. Removing the clipping costs \(CL\sqrt\tau\) by the preceding integral identity. Thus \[\left|\log|\det T|-\log|\det B|\right| \le CL\bigl(\sqrt\tau+\xi\tau^{-2}\bigr).\] Take \(\tau=\xi^{2/5}\) to obtain the first assertion.

For the second assertion choose \(0<\alpha<t_0\) and split the singular-value expansion of \(T_{SS}^{-1}\) as \(B_{\rm hi}+B_{\rm lo}\), where \[\operatorname{rank}B_{\rm hi}\le CL\sqrt\alpha, \qquad \|B_{\rm lo}\|\le\alpha^{-1}.\] This is precisely the count for singular values of \(T_{SS}\) below \(\alpha\). Every block of \(T-B\) has trace norm at most \(\xi L\), while the off-diagonal blocks have operator norm at most \(C\). It follows that \[\operatorname{Re}T_e=I+F+E,\qquad \operatorname{rank}F\le2CL\sqrt\alpha, \qquad \|E\|_1\le C\xi L(1+\alpha^{-1}),\] with \(F,E\) Hermitian. The factor two in the rank bound accounts for taking a Hermitian part. Since \(\operatorname{Re}T_e>0\), interlacing and integration of its counting function below one imply \[\operatorname{Tr}(I-\operatorname{Re}T_e)_+ \le\operatorname{rank}F+\|E\|_1.\] Indeed a rank-\(r\) perturbation changes a count by at most \(r\), and integration over \((0,1)\) then bounds the contribution of \(F\) by \(r\); the contribution of \(E\) is bounded by its trace norm. For \(0<\tau<t_0\), the clipped negative logarithm satisfies \[-\log\max(\lambda,\tau) \le\frac{\log(1/\tau)}{1-\tau}(1-\lambda)_+.\] Combining this bound with the logarithmic tail integral gives \[\sum_{\lambda<1}-\log\lambda \le2CL\sqrt\tau+ \frac{\log(1/\tau)}{1-\tau} \left(CL\sqrt\alpha+C\xi L(1+\alpha^{-1})\right).\] Taking \(\alpha=\tau=\xi^{2/3}\) proves the stated estimate. The same argument with \(\|\operatorname{Re}T-I\|_1\le CL\) and fixed \(\tau\) also gives \(\sum_{\lambda<1}-\log\lambda=O(L)\) for \(\operatorname{Re}T\). ◻

Proof. Before centering, the product equals the circle Gram determinant of \(e^f\) of size \(2n\): shift its consecutive frequencies to centered half-integers, then split into cosines and sines by evenness. The two evaluation determinants factor \(\sqrt2\cos(t/2)\) respectively \(\sqrt2\sin(t/2)\) per variable times a polynomial Vandermonde with leading coefficients \(2^j\), yielding exactly the two measures (by determinant expansion/integration).

For completeness this circle bound follows already on polynomials \(f\) and then by approximation. In the Vandermonde-squared probability on \(m\) circles (here \(m=2n\)), the centered integral is the squared norm of \(\exp(\sum_{k\ge1} f_k p_k)\) where \(p_k=\sum_{l=1}^m z_l^k\). Symmetric holomorphic polynomials have the orthonormal alternant-quotient basis (strictly increasing nonnegative monomial powers in the determinant numerator, ordinary Vandermonde denominator), by integrating determinants. Its reproducing kernel summed over all degrees, as a formal graded identity, is \(\prod_{i,j}(1-z_i\bar u_j)^{-1}\) by Cauchy–Binet then the elementary Cauchy determinant formula. This kernel equally expands using \(\exp(\sum p_k(z)\bar p_k(u)/k)\). Thus the functions \(\prod_k (p_k/\sqrt k)^{\alpha_k}/\sqrt{\alpha_k!}\) in each degree have synthesis norm at most 1 from coefficient \(\ell^2\) (matrix times its adjoint equals the identity in the orthonormal basis). The indicated exponential series hence has squared norm at most \(\exp(\sum k|f_k|^2)\); on polynomial input this series converges to the function also pointwise/absolutely, or take radial limits. This proves (36). ◻

Proof. For each label, Cauchy–Schwarz in the unnormalized integral and \(w_0=\sqrt{w_+w_-}\) bound the tilted integral by the geometric mean of the two tilted \(w_\pm\) integrals. Lemma 25 bounds that mean after subtracting \(\nu_\sigma f_{\sigma,0}\). Divide by \(J_{\nu_\sigma}(w_0)\) and multiply the two estimates. The exact normalization above gives the last inequality, including \(\nu_\sigma=0\). ◻

Proof. We compare the two Jacobi weights \(w_+\) and \(w_-\) after opposite small transports of the particles. Their geometric mean cancels the terms linear in the source and in the empirical fluctuation. The surviving gain will be \(\gamma\int(a-2a^2)+o(L)\), where \(a\) is the transport primitive. Its maximizing value \(1/4\) gives \(\gamma L/4\) per label, the logarithmic normalization loss up to the small boundary error. We now justify this comparison, including the Jacobian and its uniform error on the event.

The transport and its smooth error. Fix \(0<\delta<1/4\). Choose a smooth primitive \(a\) with \(0\le a\le1/4\), equal to \(1/4\) on \(|s|\le(1-2\delta)L\), and zero for \(|s|\ge(1-\delta)L\). It may be chosen monotone in each transition region, with \(\|a^{(r)}\|_\infty\le C_{r,\delta}L^{-r}\) for \(r\ge1\). For each label and \(\epsilon\in\{+1,-1\}\), define \[F_{\sigma,\epsilon}(s) =s-\epsilon\frac{a(s)}{\nu_\sigma\lambda_L(s)}.\] These are increasing diffeomorphisms, equal to the identity near the endpoints. For every fixed derivative order, \[\|F_{\sigma,\epsilon}-\mathrm{id}\|_{C^r} \le C_{r,\delta}L^{C_r}e^{-c_\delta L}.\] Indeed \(\nu_\sigma\lambda_L\ge c e^{\gamma\delta L}\) on the support of \(a\), and all relative derivatives of \(\lambda_L\) there are bounded. Write \(m_\sigma=\nu_\sigma\lambda_Lds\). Taylor expansion of pushforward against a test function gives \[d_{\sigma,\epsilon}:=(F_{\sigma,\epsilon})_*m_\sigma-m_\sigma =\epsilon a'\,ds+r_{\sigma,\epsilon}(s)\,ds, \qquad \|r_{\sigma,\epsilon}\|_{C^r} \le C_{r,\delta}L^{C_r}e^{-c_\delta L}.\] For example, the first-order term is \(-[\nu_\sigma\lambda_L(F_{\sigma,\epsilon}-s)]' =\epsilon a'\); every higher term contains one additional exponentially small displacement. The remainder is supported in a slightly enlarged core, has zero integral, and obeys the same bounds after integration.

We prove the comparison for one label, temporarily suppressing \(\sigma\). Put \[h(s,v)=H_{\rm ref}(\theta(s),\theta(v)),\quad K_\epsilon(s,v)=h(F_\epsilon(s),F_\epsilon(v))-h(s,v),\quad g(s)=\log\cot(\theta(s)/2).\] The ratio of the two cos-differences defining \(K_\epsilon\) extends smoothly across the diagonal. Its diagonal value is \[K_\epsilon(s,s) =-2\log\frac{\sin\theta(F_\epsilon(s))\, \theta'(F_\epsilon(s))\,F_\epsilon'(s)} {\sin\theta(s)\,\theta'(s)}.\] The one-particle base density in \(s\) is \(w_0(\theta(s))\theta'(s)/(2\pi) =\sin\theta(s)\lambda_L(s)\). Consequently its Jacobian and density ratio cancel exactly the half-diagonal contribution omitted from the Vandermonde interaction. For \(X=\sum_i\delta_{s_i}=m+q\), the logarithm of the pulled-back base-density ratio is therefore exactly \[-\frac12(X,K_\epsilon X).\] Since \(hm=0\), its mean–mean and mixed terms satisfy the identities \[(m,K_\epsilon m)=(d_\epsilon,hd_\epsilon),\qquad (q,K_\epsilon m)=(q,(hd_\epsilon)\circ F_\epsilon).\] It follows that, uniformly on \(\mathcal G\), \[-\tfrac12(X,K_\epsilon X) =-\epsilon(q,ha')-\tfrac12h_0(a,a)+o(1).\]

Here is the precise uniform error estimate. If \(q=Q'\), then \[|(q,b)|\le\|Q\|_{-2,A}\,\|b'\|_{2,A},\qquad |(q,Kq)|\le\|Q\|_{-2,A}^2 \|\partial_s\partial_vK\|_{2,A;2,A},\] where the positive Sobolev norms are dual to the negative norm in the statement, and the last norm is its tensor product. Derivatives through order three in each variable suffice to bound the last expression, up to a polynomial factor from the interval length. The divided difference argument for \(K_\epsilon\), together with the sine bounds away from the diagonal, bounds all these derivatives by \(L^Ce^{-c_\delta L}\). The kernel has even reflections at both ends, so its mixed derivatives have the odd reflections required for this Sobolev pairing. Convolution of \(h\) with the smooth core-supported remainder \(r_\epsilon\) has the same bound up to a polynomial factor: subtract the diagonal logarithm and transfer derivatives to \(r_\epsilon\), as in Lemma 19. The assumptions \(\|Q\|_{-2,A}\le L^D\) and fixed polynomial source bounds thus make every error in the preceding formula \(o(1)\). This argument also justifies the expansions that follow.

The change of the source contributes \[(f\circ F_\epsilon-f,X)=\epsilon(f,a')+o(1).\] Since \(w_\epsilon/w_0=e^{\epsilon g}\), the extra weight contributes \[\epsilon(g\circ F_\epsilon,X) =\epsilon(g,X)+(g,a')+o(1).\] Only the core derivatives of \(g\) are involved in this expansion; although \(g\) diverges at the endpoints, the map is the identity there. If \(D_\epsilon(X)\) denotes the pulled-back unnormalized density with weight \(w_\epsilon e^f\), and \(D_0(X)\) the original density with weight \(w_0e^f\), we have obtained \[\log\frac{D_\epsilon(X)}{D_0(X)} =\epsilon\bigl((g,X)+(f,a')-(q,ha')\bigr) +(g,a')-\tfrac12h_0(a,a)+o(1).\] The terms odd in \(\epsilon\) cancel in the geometric mean. Define \(G_a=(g,a')-h_0(a,a)/2\). Then on the same event \(\mathcal G\), \[D_0(X)\le e^{-G_a+o(1)} \sqrt{D_+(X)D_-(X)}.\]

The gain and the normalization. Since \(g'=-\theta'/\sin\theta\), \(-g'\) is bounded on the core and tends to \(\gamma\) under translations into either tail whose distance from the endpoints tends to infinity. Thus \[(g,a')=-\int g'a=\gamma\int a+o_\delta(L).\] On the support of \(a'\), two points in the same tail satisfy \[h(s,v)=4\gamma\min(|s|,|v|) +O\bigl(1+|\log\min(1,|s-v|)|\bigr),\] and opposite-tail values are bounded. To verify this, use \(\theta(s)\asymp e^{\gamma s}\) in the left tail and \(\pi-\theta(s)\asymp e^{-\gamma s}\) in the right tail, factor the difference of the corresponding squared distances, and take its logarithm. The additive bounded error is uniform on these transition regions. The logarithmic error has bounded integral against \(|a'(s)a'(v)|\), since \(\|a'\|_1\le C\) and \(\|a'\|_\infty\le C_\delta/L\). For an absolutely continuous \(b\) of compact support on \([0,\infty)\), \[\iint_{[0,\infty)^2}\min(r,t)b'(r)b'(t)\,dr\,dt =\int_0^\infty b(r)^2\,dr;\] this follows by writing \(\min(r,t)=\int_0^\infty \mathbf1_{u<r}\mathbf1_{u<t}\,du\). Apply it on each half-line to obtain \[h_0(a,a)=4\gamma\int a^2+O_\delta(1),\qquad G_a=\gamma\int(a-2a^2)+o_\delta(L) \ge\frac{\gamma L}{4}-C\delta L-o_\delta(L).\]

Finally apply the geometric-mean inequality simultaneously to the two labels, using one common sign \(\epsilon\); this permits the good event to couple both configurations. Cauchy–Schwarz, removal of that event from the two positive integrals, and change of variables back to their original domains give \[\mathbb E_0[\mathbf1_{\mathcal G}e^{\ell(Q)}] \le \exp\left(-2G_a+o(1) -\sum_\sigma\log J_{\nu_\sigma}(w_0) +\tfrac12\|\ell\|_*^2\right)\] by (36); the constant Fourier coefficients cancel because \(\ell\) is centered. Uniformly for \(\nu_\sigma\asymp m\), \(\log J_{\nu_\sigma}(w_0)=-\gamma L/4+O(1)\). Hence the excess over \(\|\ell\|_*^2/2\) is at most \(C\delta L+o_\delta(L)\). Taking first \(L\to\infty\), then \(\delta\downarrow0\), proves (38) with the stated uniformity. ◻

Proof. First suppose that all data are generated by finitely many physical sine tests. Put \(Q_*=-F^{-1}l\), and complete the square before discarding any positive quadratic terms: \[-V(Q)=-m_F+h_0(Q_*,Q)-\tfrac12\|Q_*\|^2 -\tfrac12h_0(Q-Q_*,(F-I)(Q-Q_*)).\] With \(J=(I-F)_+\), the last term is at most \(\tfrac12h_0(Q-Q_*,J(Q-Q_*))\). Let \(g_J\) be a real Gaussian vector of covariance \(J\). Gaussian linearization and (37) give \[\begin{align*} \mathbb E_0e^{-V(Q)} &\le e^{-m_F-\|Q_*\|^2/2} \mathbb E_{g_J}\!\left[e^{-h_0(g_J,Q_*)} \mathbb E_0 e^{h_0(Q_*+g_J,Q)}\right]\\ &\le e^{-m_F+CL}\mathbb E_{g_J}e^{\|g_J\|^2/2} =e^{-m_F+CL}\det(I-J)^{-1/2}. \end{align*}\] The cancellation of the mixed and \(Q_*\)-terms in the second line is exact. The nontrivial eigenvalues of \(I-J\) are precisely those eigenvalues of \(F\) below one, which proves the formula.

For general smooth data, truncate in the physical \(s\)-sine basis. The resulting kernels converge with enough derivatives to converge on every fixed empirical primitive. Their operators converge in trace norm, and the linear Riesz vectors converge in \(\mathcal X\); these statements follow by summing the Fourier coefficient tails and using the bounded embedding \(\mathcal X\mathrel{% \BeginAccSupp{method=hex,unicode,ActualText=21AA}% \hookrightarrow% \EndAccSupp{}% }L^2(ds)\). For sufficiently large cutoffs the gap is at least \(g/2\), so the infima converge. To control the determinant term, on any finite-dimensional negative-excess subspace compare a truncated operator to the corresponding compression of \(F\). For positive operators bounded below by \(g/2\), the integral identity for \(\log\), or differentiation along their line segment, bounds the change of log determinant by \(2g^{-1}\) times the trace-norm error. Interlacing bounds the negative log sum of a compression by that of \(F\). Fatou’s lemma now proves (39).

The coarse argument also allows the additional cost \(-t\|Q\|_{-2,A}^2\). Indeed its operator \(B_A\), defined by \(h_0(Q,B_AQ)=\|Q\|_{-2,A}^2\), has bounded norm and trace \(O(L)\): factor it through the bounded \(L^2(ds)\) embedding and sum \((1+(k/A)^2)^{-2}\). Its physical sine truncations also converge on empirical fields. Choosing \(2t\|B_A\|\le g/2\) preserves a gap.

For the sharpened estimate, take a physical sine cutoff \(K=L^B\). Smoothness gives errors smaller than any prescribed inverse power of \(L\), by choosing a sufficiently high fixed derivative order and then \(B\). Dual Sobolev estimates make the cost error \(o(1)\) on \(\|Q\|_{-2,A}\le L^D\); trace-norm error divided by the gap and norm error in the completed square are also \(o(1)\). The Riesz vectors of the first \(K\) physical sine tests have a Gram matrix with inverse-polynomial lower bound. To verify this, test the linear combination of their functionals on the same sine combination as a primitive. Its \(L^2\)-pairing is the Euclidean squared norm, whereas the local derivative and boundary estimate for \(h_0\) bounds its \(\mathcal X\)-norm by a polynomial times that norm. Duality gives the claimed lower bound. Consequently every polynomially bounded vector in this finite-dimensional space has a smooth dual test with polynomial derivative bounds.

Apply (38) in the displayed Gaussian argument when \(\|g_J\|\le L^{B'}\). Outside this ball use (37). After the factor \(e^{\|g_J\|^2/2}\), the normalized tilted Gaussian has covariance \(J(I-J)^{-1}\), of polynomial norm, in polynomial dimension. For covariance norm at most \(L^c\) and dimension at most \(L^c\), its tail outside radius \(L^{B'}\) is bounded, for large \(B'\), by \(\exp(-L^{2B'-c}/4)\) times an exponential of a fixed polynomial. Choose \(B'\) to absorb the factor \(e^{CL}\). This gives the stated \(o(L)\) loss. ◻

Proof. We first construct an affine path of smooth fields with a positive real Hessian. We then implement this path by particle displacements, compute the compensated cost and the remaining Jacobian, and estimate the region where a cutoff switches off the deformation.

Proof of Proposition 29. For the estimates without a shift, take absolute values, use \(0\le A\le1\), and apply Lemma 28 with \(F=\operatorname{Re}T\) and infimum \(m_0\). The negative log sum is \(O(L)\) by (32) and the trace bound; the reference masses are bounded above. The real trial estimate and (29), with (27), prove the first two assertions.

On the full-shift range use the sharp version of (39), (35), and the infimum comparison to \(m_z\). Including the full Jacobian and both \(J_{\nu_\sigma}(w_0)\) gives the log bound there \[-m_z+(7\gamma/8)L-\gamma L/2+o_\xi(L)+o(L)\] by (33). Taking \(\xi\) small, then applying the deterministic lower bounds on \(m_z\), and the separate negligible bad-region estimate (choose its power after the polynomial restrictions), proves (41) and (42). ◻

Proof. Use the Gaussian probability measure \(d\mu_a=(a/\pi)^{d/2}e^{-a|y|^2}dy\), and its orthonormal Hermite polynomials defined by \[e^{\sqrt{2a}z^{\mathsf t}y-z^{\mathsf t}z/2} =\sum_\alpha H_\alpha(y)\frac{z^\alpha}{\sqrt{\alpha!}}.\] Gaussian integration proves orthogonality. For completeness, polynomials are dense: an orthogonal complement would have an entire Gaussian-weighted Laplace transform with every derivative at zero equal to zero; Fourier uniqueness then makes the complement zero. The integrability needed here follows from Gaussian decay.

Let \(c_\alpha=\int gH_\alpha\,d\mu_a\). Completing the square gives \[\begin{align*} \sum_\alpha c_\alpha\frac{z^\alpha}{\sqrt{\alpha!}} &=e^{-c_1}\det(I+D/(2a))^{-1/2}\\ &\quad\times\exp\!\left\{ \tfrac12l^{\mathsf t}(2aI+D)^{-1}l -\sqrt{2a}z^{\mathsf t}(2aI+D)^{-1}l -\tfrac12z^{\mathsf t}D(2aI+D)^{-1}z\right\}. \end{align*}\] The square root is the analytic branch specified by the Gaussian integral. Accretivity ensures its existence and polynomial inverse norms. Set \(B=D(2aI+D)^{-1}\). The identity \[(2aI+D)^*(2aI+D)-D^*D =4a(aI+\operatorname{Re}D)\succeq2a\delta I\] shows \(\|B\|\le1-L^{-C}\). The other factors in the generating function have size at most \(\exp(L^C)\). Therefore, for some \(r=1+L^{-C_1}>1\), integrating the squared generating function at \(rz\) against \(\pi^{-d}e^{-|z|^2}\,dz\) on \(\mathbb C^d\) gives \[\sum_\alpha r^{2|\alpha|}|c_\alpha|^2\le e^{L^C}.\] This follows from monomial orthogonality; convergence follows from \(r^2\|B\|<1\), with inverse-polynomial margin.

The Hermite generating identity also gives Mehler’s diagonal formula [17] \[\sum_\alpha t^{|\alpha|}|H_\alpha(y)|^2 =(1-t^2)^{-d/2} \exp\!\left(\frac{2at}{1+t}|y|^2\right),\quad 0<t<1.\] One direct derivation writes each one-variable unnormalized Hermite polynomial as the moment of \(\sqrt{2a}y+iG\), for a standard real Gaussian \(G\), and integrates the summed exponential for two independent copies. Taking \(t=r^{-1}\) bounds the right side by \((1-r^{-2})^{-d/2}e^{a|y|^2}\). Cauchy–Schwarz now bounds the Hermite tail above degree \(K\), after multiplying by \(e^{-a|y|^2/2}\), by \(e^{L^C}r^{-K/2}\). The prefactor includes the polynomial-dimensional Mehler factor. A polynomially large \(K=K_1\) attains any specified \(B_1\). Completeness and local uniform convergence identify the series with \(g\). Expanding the retained Hermite polynomials in monomials gives the asserted coefficient bound, since the dimension and degree are polynomial. The Gaussian coefficient integrals, uniformly dominated on compact \(M\)-sets, prove holomorphy. ◻

Proof of Proposition 31. Use the polynomial and endpoint-subset approximations just constructed. In each retained subset integral, replace the higher cumulants and kernel corrections by their continued near-index formulas, the constant kernel by the continued Christoffel–Darboux expression, and the linear traces and masses by their polynomial-antidifference approximants. Reconstruct \(\mathcal D_n\) by formal exponentiation through source degree \(K_1\), and take the indicated source derivatives. The two labels are combined by independence.

Every expression before endpoint integration is holomorphic in \(\nu,M\), with a uniform bound \(e^{L^C}\). Indeed source types, source orders, and determinant sizes are polynomial. Near-index blocks have length \(O(m)\), so an expression with polynomially many index sums has at most \(\exp(\operatorname{poly}(L))\) terms. Permutation expansions, source coefficient extractions, and factorials of polynomial orders satisfy the same bound. Coefficients \(c_\alpha\) include only the fixed polynomial \(P\), centerings, and bounded endpoint data. Integration over fixed real endpoint regions preserves these bounds and holomorphy by dominated convergence (or Morera’s theorem). No analyticity of the endpoint factors themselves as functions of their real positions is needed.

We record the order of the error choices to exclude a circular precision requirement. First fix \(B_0\) using the original integer-law expectation bound. This fixes the dimension and gap in the Gaussian approximation. Next fix \(B_1\) and \(K_1\); then \(D_1,B_2,D_E\), in that order; finally choose \(K_e\) for the factorial tail. All errors to this point have already been bounded on the actual integer laws by \(e^{-L^b}\), with a freely reserved margin. They are not passed through an analytic continuation formula. The remaining formulas, through degree \(K_1\), are polynomial in the linear trace coefficients and in the kernel entries. Their sensitivity to the final trace and normalization errors is at most \(e^{L^{C_*}}\), for an exponent \(C_*\) depending only on the choices already made. The preceding size bounds for trace and mass approximants are independent of their final polynomial Taylor order. Choose \(B_3>\max(b,C_*)+1\), and the corresponding orders, last. The resulting errors are negligible after this final amplification. Taking the initially reserved margins, for example replacing each requested error by \(e^{-2L^b}\), makes their finite sum at most \(e^{-L^b}\) for large \(L\). This proves both assertions in (44). ◻

Proof. Taylor-expand \(\widehat F\) in its two dimension variables at \((m,m)\). Its bound \(e^{L^C}\) on the radius-\(\sqrt m\) polydisk implies that a polynomial \(P\) of polynomial degree \(K\) approximates it on the half-radius polydisk to an error \(o(B)\), by choosing \(K\) so that \(e^{L^C}2^{-K}=o(B)\). Put \(R=\lfloor\sqrt m/3\rfloor\), and let \(S\) be the sup norm of \(P\) on the real square \([m-R,m+R]^2\). The rescaled one-variable Chebyshev expansion has coefficients at most twice its sup norm and \(\|T_k'\|_{[-1,1]}\le k^2\). Holding the other variable real therefore bounds each partial derivative by \(CK^3S/R\). Comparing a maximizer with a nearest integer point gives \[S\le \max_{\text{integer square}}|P|+CK^3S/R.\] Since \(R\) is exponential and \(K\) polynomial, the second coefficient is \(o(1)\). Proposition 31 and \(b>b'\) then give \(S\le(1+o(1))B\).

The tensor Chebyshev coefficients are at most \(4S\), and there are at most \((K+1)^2\) of them. At the target, each rescaled coordinate has magnitude \(\operatorname{poly}(L)/R\). Its complex inverse-cosine angle differs from \(\pi/2\) by the same order, so every Chebyshev basis value up to degree \(K\) is bounded by \(\exp(CK\operatorname{poly}(L)/R)=1+o(1)\). Thus \(|P(\nu^*)|\le C(K+1)^2B\), which is \(B\exp(o(L))\). The Taylor error is negligible relative to \(B\). ◻

Proof. The common eigenvector is independent of \(z\). Computing the empty output of the transfer on that vector proves the rationality and denominator bound. A shift \(z\mathrel{% \BeginAccSupp{method=hex,unicode,ActualText=21A6}% \mapsto% \EndAccSupp{}% }z+\pi\) inserts occupancy signs that cancel around the closed auxiliary. At either imaginary infinity, \(A,U\to0\), \(B\to1\), and all entries remain bounded. Only empty input reaches empty output; it contributes the empty diagram and one essential auxiliary circle, with weights \(1\) and \(\chi\). Reflection reverses the row order and negates all \(b_i\); the balanced list is thereby preserved, and simplicity identifies the reflected eigenline with the original one.

For the pinch, pass the two auxiliaries in order \([u+j\lambda,u]\). At a host with \(x=u-b_k\) their crossing is \[M_x=R_1(x+j\lambda)R_2(x),\qquad (I\otimes S_j^{\mathsf t})M_x =\widehat R(S_j^{\mathsf t}\otimes I).\] At the marked host, \(x=0\) and \(M_0=(S_j\otimes I)(S_j^{\mathsf t}\otimes I)\). Close the paired auxiliary trace and move the transpose splitter backwards through all other hosts. At the marked host the remaining contraction is \[(I\otimes S_j^{\mathsf t})(S_j\otimes I) =\widehat R\big|_{x=0}.\] For completeness, this identity follows by multiplying on the right by \(S_j^{\mathsf t}\otimes I\), using the transposed splitting identity at \(x=0\), and cancelling that factor. Cancellation is valid because \(S_j^{\mathsf t}\) is surjective: insert vacancy, or one through strand into a fixed output and divide by its nonzero coefficient. The fused auxiliary is the one of argument \(u+\lambda\) for \(j=2\), and disappears for \(j=3\). These are identities of annular diagrams, so they prove (53) with the essential-loop fugacity retained.

The first two derivatives of \(\log\Lambda\) are \(f\) and \(g\). Taking their derivatives in (53) makes \(\mathscr L f\) and \(\mathscr L g\) vanish at the points corresponding to both \(b_i+\lambda\) and \(b_i+2\lambda\). At generic sites the factors giving these zeros are distinct. Analyticity of the normalized eigenvector near the balanced physical sites, and regularity of the boxes at all the evaluation points, permit confluence. The two families give the stated multiplicities. The two sine factors in each \(\mathscr D(z-b_i)\) give the pole list. Since the denominator is independent of \(\chi\), taking two \(\chi\)-derivatives does not increase those orders. ◻

Proof. We first solve the jet equation in the Hilbert norm. We then obtain the fourth-norm estimate needed for the quadratic term \(f^2\), and divide out the prescribed zeros to estimate the residual.

Proof. The residual estimate already gives both upper bounds. The lower bound requires a separate argument: after shifting to the edge \(A_N=\gamma^{-1}\log(3N)\), we identify a limiting interpolation problem and show that its contribution at the center is strictly positive.

Proof. In a slab of \(M\) columns with vacant outer cuts, differentiate the logarithm of the loop partition at \(\chi=0\). Its first derivative counts single essential polygons. Its second derivative is minus the weight of ordered incompatible pairs, including a polygon paired with itself. After division by \(M\), strict spectral dominance gives convergence of the two derivatives to those of \(\log\Lambda(\chi;z_0)\): the spectral projection has nonzero vacuum overlap, equal to one at zero fugacity, and the remaining spectrum decays exponentially, locally uniformly in the analytic parameter. Combinatorially each finite polygon or intersecting pair occupies a finite consecutive interval of columns; its number of placements divided by \(M\) increases to one. Monotone convergence gives \[ \begin{split} f(0)&=\sum_{[\ell]}W(\ell),\\ -g(0)&=\sum_{[\ell],[\ell']}W(\ell)W(\ell') \sum_{j\in\mathbb Z} \mathbf1_{\{\ell\cap(\ell'+j)\ne\varnothing\}}. \end{split} \tag{78}\] The physical tiles encode precisely the honeycomb polygons with their vertex weights, and on this embedded trivalent graph intersection is precisely incompatibility.

Let \(\Gamma\) be the triangular translation lattice modulo \(V\). Its integer horizontal subgroup has index \(N\). Fix vertices \(P,Q\) of \(\ell\) of the same bipartite type, with \(\Delta_x=x(Q)-x(P)>0\), and put \(\Delta=Q-P\in\Gamma\). For a translating vector \(a\in\Gamma\), let \(I_P(a)\) indicate that \(P\) lies strictly in the right-hand component of the complement of \(\ell'+a\). Along each horizontal coset it is one on the negative tail and zero on the positive tail. Moreover \(I_Q(a)=I_P(a-\Delta)\). The difference is finitely supported, and \[\sum_{a\in\Gamma}(I_Q(a)-I_P(a))=N\Delta_x.\] For an integer horizontal \(\Delta\), this is ordinary telescoping in each of the \(N\) cosets. In general multiply \(\Delta\) by \(N\), which is an integer horizontal translation modulo \(V\), and telescope the sum of the \(N\) translates of the original difference. If the indicators disagree, the path in \(\ell\) from \(P\) to \(Q\) meets \(\ell'+a\); a point on its boundary also gives intersection. Thus at least \(N\Delta_x\) translating vectors give an intersection. Choosing the vertices near the two horizontal extremes, and changing one to a neighbor when necessary to match bipartite types, gives \(\Delta_x\ge d_x(\ell)-C\).

Average the inner intersection count in (78) over representatives of \(\Gamma/\mathbb Z\). Translation by each representative permutes the horizontal polygon classes and preserves their weights. The resulting inequality is \[f(0)\sum_{[\ell]}W(\ell)(d_x(\ell)-C)\le-g(0).\] Now apply Lemma 39; in particular \(-g(0)\ge0\) by (78). This proves (77).

Every separating essential polygon meets a fixed arc joining the two markers. That arc has horizontal span \(O(N)\). A polygon class can therefore have at most \(CN+d_x(\ell)+C\) translates meeting it. The two estimates in (77) bound their total weight. If this bound is \(C\), summing all sets of removed polygons without any compatibility restrictions costs at most \(\sum_{k\ge0}(\chi C)^k/k!=e^{\chi C}\). Positivity provides the opposite inequality by retaining only configurations with no removed polygon. ◻

Proof. For unitarity, the one-occupancy sector is the two-by-two matrix with diagonal \(A\) and off-diagonal \(B\). Direct substitution in (79) gives its inverse after normalization by \(L_yL_{-y}\). In the even-occupancy sectors the required identities are \[\begin{gathered} e_ye_{-y}+nU_yU_{-y}=E_yE_{-y}=L_yL_{-y},\\ e_{-y}U_y+(E_y+nF_y)U_{-y}=0,\\ E_yF_{-y}+F_yE_{-y}+nF_yF_{-y}+U_yU_{-y}=0. \end{gathered}\] They prove \(R_yR_{-y}=I\). The factorization at \(j\lambda\) follows by substituting \(y=2\lambda,3\lambda\).

We record the coefficient check for splitting to retain the terms arising from the nonzero cycle weight. In the following tables, subscript \(1\) means argument \(x+j\lambda\), subscript \(2\) means argument \(x\), and every letter has been divided by the corresponding \(e\). Unsubscripted letters belong to \(\widehat P\); for deletion take \(B=1\) and the other nonempty two-slot coefficients zero. Write \(q=q_j\) and \(k=q_2\) for \(j=2\), \(k=0\) for \(j=3\). Outputs are numbered \(1,2,3\), with \(1,2\) the split slots; a parenthesized pair denotes an additional cup. In the last block a parenthesized output pairs the two input strands, whereas an unparenthesized pair denotes through strands. The coefficient equalities are \[\begin{array}{c|c|c|c} \text{input}&\text{output}&\text{two boxes}&\text{fused value}\\\hline 00&\varnothing&d_j&d_j\\ &12&U_1A_2+qB_1B_2&d_jq\\ &13&U_1B_2+qB_1A_2&d_jkU\\ &23&U_2+qA_1(E_2+nF_2)&d_jkU\\\hline 10&1&A_1+q(F_1+nE_1)U_2&d_jkA\\ &2&B_1A_2+qU_1B_2&d_jkA\\ &3&B_1B_2+qU_1A_2&d_jB\\ &3,(12)&qF_1E_2&d_jqB\\ &1,(23)&A_1U_2+q(E_1E_2+F_1F_2+nE_1F_2)&0 \end{array}\] and, for \(j=2\), \[\begin{array}{c|c|c|c} 01&1&k(B_1+U_1U_2)&d_2kB\\ &2&k(A_1A_2+B_2)&d_2kB\\ &3&k(A_1B_2+A_2)&d_2A\\ &1,(23)&k(B_1U_2+U_1F_2)&0\\ &3,(12)&kU_1E_2&d_2qA\\\hline 11&\varnothing&k(U_1+B_1U_2)&d_2U\\ &12&k(E_1A_2+A_1B_2)&0\\ &(12)&kF_1A_2&d_2qU\\ &13&k(E_1B_2+A_1A_2)&d_2kE\\ &(13)&kF_1B_2&d_2kF\\ &23&kB_1E_2&d_2kE\\ &(23)&k(U_1U_2+B_1F_2)&d_2kF \end{array}\] The terms proportional to \(n\) are exactly the closed cups in the contraction. These tables can be verified without diagrammatic simplifications: put \([s]=T^s-T^{-s}\), \(T=e^{i\lambda}\), and \(p=x/\lambda\). At argument \(x+l\lambda\), the normalized tuple \((A,B,U,E,F)\) is \[\frac{([2][3-p-l],\,[p+l][3-p-l],\,[2][p+l],\, [3-p-l][2-p-l],\,[p+l-1][p+l])} {[2][3]+[p+l][3-p-l]}.\] Substitute \(q_2=[1]/[2]\), \(n=-(T^4+T^{-4})\), and clear denominators. Multiplication gives the displayed right columns, proving all occupancy sectors of (80).

For the three-box identity, specialize its second argument to \(y=j\lambda\), \(j=2,3\). Splitting, transposition, and reflection reduce both sides to the same fused diagram with scalar \(d_j(x)\). Unitarity gives the further specializations \(y=j\lambda-x\); \(y=0\) follows from \(R_0=I\). After clearing scalar denominators, each occupancy component has exponents between \(-4\) and \(4\) in \(e^{iy}\), all of fixed parity. The five values modulo \(\pi\) just listed therefore determine the polynomial at generic \(x\), and rational continuation proves the identity everywhere it is defined. ◻

Proof. Exchanging adjacent auxiliaries \(v,u\) uses the crossing \(R_{v-u}\) at the bottom, moved to the top by the three-box identity. For exchanging the two lines of one transfer, the boundary identity is \[R_{\bar z-z}K(\mathfrak m(\bar z)) \ \text{is proportional to}\ K(\mathfrak m(z)).\] The proportionality follows from, at \(y=\bar z-z\), \[U_y+\mathfrak m(\bar z)(E_y+nF_y) =\mathfrak m(z)(e_y+n\mathfrak m(\bar z)U_y).\] The transposed inverse crossing contributes the reciprocal scalar, proving reflection symmetry.

For two pairs, the exchanges taking \([\bar v,v,\bar u,u]\) to \([\bar u,u,\bar v,v]\) have operator product \[(R_{-w_0})_2(R_h)_3(R_{-h})_1(R_{w_0})_2, \qquad h=v-u,\quad w_0=v+u-3\lambda,\] with the rightmost factor acting first. They take the product of bottom caps exactly to its reversal. By unitarity, this reduces to left-right symmetry of \((r(h))_3(r(w_0))_2[K(a)\otimes K(b)]\), where \(a=\mathfrak m(\bar v)\), \(b=\mathfrak m(\bar u)\). Only the paired output types \(12,34\) and \(13,24\) need comparison; their equalities are \[ \begin{aligned} aA_{w_0}e_h+ab(nE_{w_0}+F_{w_0})U_h &=e_{w_0}U_h+bA_{w_0}(E_h+nF_h),\\ (a-b)B_{w_0}A_h&=(1-ab)U_{w_0}B_h. \end{aligned} \tag{83}\] Substitution from (79) and the sine formula for \(\mathfrak m\) proves them. Replacing \(v,u\) by \(\bar u,\bar v\) and transposing gives the top exchange, hence commutation.

At the physical center both cap coefficients vanish. Spectral simplicity of the open-strip transfer gives an analytic simple eigenspace nearby, invariant under all \(t(z)\) by commutation. The empty-coordinate rule at \(n=0\) makes its eigenvalue identically one. At imaginary infinity, \(A_y/e_y,U_y/e_y\to0\), \(B_y/e_y\to1\), and \(\mathfrak m(z)\mathfrak m(\bar z)\to1\). Thus the empty-output computation is \(1+n\). Occupancy signs prove periodicity, and computing on the fixed common eigenvector proves rationality.

For the pinch, exchange the middle lines using \(R_{w_0}\) on the bottom caps and \(R_{-w_0}\) on the top, reaching order \([\bar v,\bar u,v,u]\). The descending pair \(v,u\) crosses first. At host \(x_i=u\), the factorization of \(P_{j\lambda}\), together with \(P_0=I\), inserts a transpose splitter at input. Move it backwards through the other hosts by the transposed version of (80). Its contraction against the exchanged bottom caps splits the first pair by \(S_j\), with scalar \(c_b\). In raw coefficients the necessary identities are \[ \begin{aligned} aA_{w_0}+q_jab(nE_{w_0}+F_{w_0}) &=q_j(e_{w_0}+nq_jbA_{w_0}),\\ aB_{w_0}+abU_{w_0} &=U_{w_0}+bB_{w_0}\\ &=\mathfrak m(\bar u-\lambda) (e_{w_0}+nq_jbA_{w_0})\quad(j=2). \end{aligned} \tag{84}\] Here \(a=\sin(v-3\lambda/2)/\sin(9\lambda/2-v)\), and likewise for \(b\); multiplying out proves (84), just as for (83). Move the first pair’s splitter through the hosts by the reflected splitting identity. The reflected top calculation, with \(\bar u,\bar v\), gives \(c_t\). The remaining descending pair has both an input and output transpose splitter; transposed intertwining reduces it to the other fused transfer. Cancelling the surjective input splitter is valid as in the annular proof. Every crossed host contributes the two factors displayed in (81). This proves that identity for generic other sites, hence by continuation. ◻

Proof. The poles of \(\mathcal F_o-1\) determine the space of unknowns, whereas zeros of \(Y\) determine the interpolation equations. These spaces have the same dimension but different two-dimensional additions to their common base. We first show that the resulting interpolation remains uniformly invertible. We then remove the bulk forcing \(N\phi\), treat the two boundary poles of \(b\), and estimate what remains.

Proof. At the moving center \(z=3\lambda/2\), and with the hosts also taken at \(\lambda/2\), both auxiliary caps are vacancies for all nearby \(\lambda\). The local vacuum weights for \(P_y\) are identically one. Differentiating the finite-strip partition at \(n=0\) therefore counts exactly one polygon, with its original critical honeycomb weight: derivatives of local occupied weights have an additional factor \(n\) and disappear at zero. Changing the arguments with \(\lambda\) contributes nothing to the eigenvalue derivative, since the base eigenvalue is constant.

The transfer \(t\) advances two horizontal columns. Spectral dominance, as in (78), identifies \(\mathcal F_o(0)\) with twice the polygon density per single column. Normalize a representative of a full translation class so that its lowest occupied band is zero. Its lowest band can then be placed at precisely \(0,1,\ldots,N-H(P)-1\) in the strip. Horizontal half-unit offsets caused by a one-band translation do not affect density: for each of these \((N-H(P))_+\) placements, integer horizontal translates have density one per column. Monotone counting of finite longitudinal placements thus gives \(Q_N=\mathcal F_o(0)/2\). All passages from finite length use nonnegative polygon sums and the analytic finite-dimensional transfer; they require no interchange of an uncontrolled infinite derivative. Equation (90) proves (97), with \[c_1=\tfrac12(\rho_* *\phi)(0),\qquad c_2=\tfrac12(\rho_* *b)(0).\] ◻

Proof. Divide (97) by \(N\). Since \((1-H(P)/N)_+\) increases to one, monotone convergence gives \(\sum_{[P]}\rho^{|P|}=c_1<\infty\). Define \[D_N=c_1N-Q_N=\sum_{[P]}\rho^{|P|}\min(N,H(P)).\] This sequence is increasing, and \(D_N=-c_2+O(N^{-1})\). Choose an admissible integer \(M\asymp N/2\). Every polygon with \(H(P)>N\) contributes \(N-M\) to \(\min(N,H(P))-\min(M,H(P))\), so \[(N-M)\sum_{[P]:H(P)>N}\rho^{|P|} \le D_N-D_M=O(N^{-1}).\] Thus the height tail is \(O(N^{-2})\). This argument also works if the pressure formula is used only along even widths, by taking \(M,N\) even. Rotate the lattice to obtain the same bound in the three band directions. The Euclidean diameter is bounded by a constant times the largest of these three band spans, plus a fixed lattice constant. The union bound proves (98) for large \(R\); total summability covers bounded \(R\). ◻

Proof. For a planar shape of diameter comparable to \(R\), the number of translations fitting in the strip and enclosing the specified point is at most \(CN(1+R)\): there are \(O(N)\) possible vertical placements, and enclosure forces its horizontal range to contain the point. For dyadic \(R\ge c_0N\), Theorem 45 bounds the weight of all such shapes by \(CR^{-2}\). Thus their total weight is bounded by \[C\sum_{j\ge0}N(1+2^jc_0N)(1+2^jc_0N)^{-2} \le C_{c_0}.\] A simple contractible cylinder polygon lifts to translates of a simple planar polygon with disjoint interior disks. The boundary lifts are disjoint; if two of their bounded disks overlapped, planar separation would force nesting, which is impossible for distinct translates of equal-area bounded disks. Thus its disk projects injectively to the cylinder. For a fixed planar translation class, the number of placements modulo the period containing the marked point is again at most \(CN(1+R)\): there are \(N\) vertical translation classes modulo period and its horizontal range must straddle the marker. Counting all these placements, including those not valid under projection, is an upper bound. The same dyadic estimate applies.

For a fixed remaining nest, sum the removed polygons while ignoring disjointness both among themselves and with the nest. If their total single-polygon weight is at most \(C_{c_0}\), this contributes at most \(e^{\chi C_{c_0}}\). Positivity gives the reverse inequality by taking the empty removed set. ◻

Proof. The two cylinder markers have disjoint neighborhoods of radius \(c_0N\), each isometric to its planar neighborhood, if \(c_0\) is small enough: along the standard balanced column cut their displacement is \(-N/4+iN\sqrt3/4\), and their vertical separation is half the balanced period. Figure 3 shows the two embedded neighborhoods. Delete all separating essential polygons, using Proposition 40, and all contractible polygons of lifted diameter greater than \(c_0N\), using Lemma 46 at both markers. The product of the deletion costs is uniformly bounded.

Every remaining polygon is contractible and encloses exactly one marker in its disk. A planar simple curve of diameter at most \(r\) enclosing a point lies in the radius-\(r\) neighborhood of that point: the point is in the curve’s convex hull, so its distance to each curve point is at most the curve’s diameter. Hence each remaining polygon lies in its marker’s embedded neighborhood. Conversely, a planar nest counted by \(P_\chi(c_0N)\) projects into that neighborhood and separates the cylinder markers. The neighborhoods are disjoint, so the two nests can be chosen independently. Retaining just these configurations gives the lower bound, and the preceding deletion gives the upper bound in (99).

For each fixed \(\chi>0\), Theorem 1 gives \(Z_N(\chi)=N^{\tau(\chi)+o(1)}\) along even \(N\). Taking logarithms in the comparison yields \(P_\chi(c_0N)=N^{\tau(\chi)/2+o(1)}\). Monotonicity in the radius and bracketing an arbitrary radius between consecutive admissible values prove (100). At \(\chi=2\), \(\tau(2)=1/6\), giving (101). ◻

Proof. Choose a fixed \(c_0>0\) so small that every nest enclosing \(f\), with each polygon of diameter at most \(c_0k\), fits inside the strip. By Lemma 46, deleting all larger polygons has a bounded cost. Therefore \[P_2(c_0k)\le S_k\le C P_2(c_0k).\] Apply Theorem 47. ◻

Proof. Cut the edge at \(p\) into two stopping outlets. For a boundary source \(a\), the winding splitting identity still has total one: every collision stem approaches its closing cycle from outside, so the two orientations cancel. Subtract this identity from the one without the cut. The total winding weight of paths visiting \(p\) equals that of paths from \(a\) stopping at the cut. Reversing these paths and taking real parts gives \(L_p(D)=F_+(D)+F_-(D)\), where \(F_\sigma\) is the real winding-weighted sum of paths from the specified side \(p^\sigma\) to the exterior. The initial direction points into the neighboring vertex on that side.

Apply the same splitting procedure from this interior source, retaining all stopping possibilities. A return to the other side of the cut traces a polygon \(\ell'\) through \(p\). For a fixed initial side it is counted once and has turning \(\pm2\pi\), hence real factor \(\cos(2\pi t)=-d\). A collision with a polygon not enclosing \(p\) cancels with its reversed orientation. If its polygon \(\ell\) strictly encloses \(p\), the stem approaches from inside. The closing turn of the counterclockwise polygon at the collision vertex is \(-\pi/3\), whereas the turn from the stem into that orientation is \(+\pi/3\). The added winding is therefore \(8\pi/3\), or \(-8\pi/3\) for the other orientation. Both have factor \(\exp(\pm it8\pi/3)=-1\).

For a fixed enclosing \(\ell\), sum the stems in its interior, excluding its vertices. This is exactly the corresponding \(F_\sigma\) for the smaller domain: the stem stops at the mid-edge before its aimed-at vertex, whose weight belongs to \(\ell\). If its initial direction already points at an excluded vertex, this stem sum is one. No further inner polygon is then possible. Write \(D_\ell\) for the domain strictly inside an enclosing polygon \(\ell\), with its vertices excluded and with stopping outlets on the incident half-edges. Conservation of the initial mass, followed by the collision classification above, gives the recurrence \[F_\sigma(D)=1+d\sum_{\ell'\ni p}\rho^{|\ell'|} +2\sum_{\ell\text{ enclosing }p}\rho^{|\ell|}F_\sigma(D_\ell).\] The factor two in the last sum comes from the two orientations of the enclosing polygon, each with real factor \(-1\) on the stopped side of the conservation identity. The initial-source convention just specified gives \(F_\sigma(D_\ell)=1\) if its first vertex is excluded.

Iterate this recurrence, choosing an outermost enclosing polygon first and then polygons strictly inside it. Every nested family is obtained once in this order. The iteration terminates either with the constant term one or with a polygon through \(p\); such a polygon must be the innermost selected cycle: \(p\) lies on its boundary, so no disjoint cycle strictly inside its bounded disk can enclose \(p\). Every enclosing cycle removes vertices, so the iteration is finite. It gives, for either sign, \[F_\sigma(D)=\mathcal Z_p(D)+d\sum_{\ell'\ni p} \rho^{|\ell'|}\mathcal Z_{\mathrm{around}}(\ell'),\] where \(\mathcal Z_{\mathrm{around}}(\ell')\) sums the disjoint outer polygons surrounding \(\ell'\), with fugacity two per polygon. This recursion remains valid if the opposite neighbor of \(p\) is excluded; in that case there is no return to the cut.

A polygon enclosing exactly one of \(f_1,f_2\) uses their separating edge. There can be at most one such polygon in a disjoint nest, and it is the innermost polygon of that nest: any smaller polygon around the same face would also have to use that edge. Consequently \[\mathcal Z_{f_1}(D)+\mathcal Z_{f_2}(D) =2\mathcal Z_p(D)+2\sum_{\ell'\ni p} \rho^{|\ell'|}\mathcal Z_{\mathrm{around}}(\ell').\] Together with the preceding expression for \(F_++F_-\) and \(0<d<1\), this proves the finite-domain claim.

Exhaust an infinite strip by parallelograms. The top/bottom path sums and the nesting sums increase. The terms with a distant side-wall endpoint tend to zero. To justify the latter assertion, split a path at \(p\) and discard avoidance between its arms. Keep a fixed group of columns containing \(p\). On either side, the vacuum transfer supplies a uniformly bounded half-state. A top/bottom endpoint is bounded by the sum over its source column followed by one-defect propagation to the fixed columns; this sum converges by the fixed-height transfer decay. A side-wall endpoint contributes a decaying one-defect power, with only finitely many wall ports. Thus each arm is uniformly summable and its side-wall part tends to zero. The upper bound on \(L_p\) obtained in this way and the lower finite-domain inequality also prove finiteness of the limiting nesting sums. ◻

Proof. Sum \(L_p(\mathcal S_N)\) over one representative of each internal mid-edge orbit under integer horizontal translation. For each summand, translate its source port to the fixed representative of its boundary orbit. The marked internal edge then runs through all visits of the translated path. This is a bijection between the two sets of path–edge pairs, preserving both length and winding; hence translation invariance turns the sum into the winding-weighted number of internal edge visits for paths from the chosen source representatives. There is one bottom-port orbit and one top-port orbit, whose source totals agree by a half-turn. Thus, if \(\mathcal E_N\) is the chosen set of internal mid-edge representatives, the orbit-count identity is \[\sum_{p\in\mathcal E_N}L_p(\mathcal S_N) =2\sum_{\gamma\in\mathcal P_N(a)} \rho^{n(\gamma)}(n(\gamma)-1)\cos(tW(\gamma)).\] A path with \(n\) vertices has \(n-1\) internal mid-edge visits, and its winding cosine is between \(c\) and one. There are \(O(N)\) internal edge orbits. These positive sums may be rearranged by monotone convergence.

Every face-center nesting mass in the strip is at most \(S_{N+1}\): embed the strip in a \(2(N+1)\)-band strip whose middle line contains the chosen face center. Lemma 49 now bounds the sum of internal edge visits by \(CNS_{N+1}\). To replace \(n-1\) by \(n\), add the boundary partition mass \(\mathcal A_N+\mathcal B_N\le 1/c\), using the strip identity \(c\mathcal A_N+\mathcal B_N=1\). This proves the first inequality; Corollary 48 proves its asymptotic form. ◻

Proof. Put \(k=\lfloor N/4\rfloor\), and take \(N\) large enough that \(k\ge1\). A path is confined to the bottom \(k\)-band strip exactly when it stays below its top line: crossing that line uses a port between the two adjacent bands. Thus the conditioning event consists of all bridges and precisely the bottom arches not counted by \(\mathcal A_k\). Its unnormalized mass is \[q_N=\mathcal B_N+\mathcal A_N-\mathcal A_k =\mathcal B_k/c-(1/c-1)\mathcal B_N \asymp N^{-1/4}.\] Indeed monotonicity gives \(\mathcal B_k\le q_N\le\mathcal B_k/c\), and the strip theorem gives \(\mathcal B_k\asymp N^{-1/4}\).

Restrict the orbit sum in the preceding proof to internal mid-edges whose heights lie in the central third. For sufficiently large \(N\), each adjacent face center is at distance at least \(N/3-O(1)>k\) bands from either boundary. Its centered \(2k\)-band strip therefore fits inside \(\mathcal S_N\), so its nesting mass is at least \(S_k\). There are at least \(c_1N\) such internal edge orbits, and Lemma 49 gives a lower bound \(c_2NS_k\) for their total winding-weighted visit mass. Any bottom-source path counted there reaches the top boundary of the bottom \(k\) bands; the rotated statement holds for a top source. Since the winding cosine is at most one and the number of counted visits is at most the path length, the unnormalized first-length mass of the conditioning event is at least \(c_3NS_k\). Its upper bound is \(\mathcal M_N(a)\). By Corollary 48 and Proposition 50, these bounds both have exponent \(13/12\). Division by \(q_N\) gives \(N^{13/12+1/4+o(1)}=N^{4/3+o(1)}\). ◻