Proof. The density of \(\alpha\) relative to its product one-site prior is proportional to \(Z_L(\alpha)Z_R(\alpha)\). For either factor, denote its bond observables by \(O_e\) and its couplings by \(J_e(\alpha)\). At two distinct vertices, \[\partial_x\partial_y\log Z =\sum_e\mathbb E(O_e)\,\partial_x\partial_yJ_e +\sum_{e,f}\operatorname{Cov}(O_e,O_f) \partial_xJ_e\,\partial_yJ_f.\] All first coupling derivatives in the left plane are nonpositive; all those in the right plane are nonnegative. A nonzero mixed derivative at distinct vertices occurs only at a bond’s endpoints and is nonnegative. The bond means and covariances are nonnegative by (9). Thus the logarithmic density has nonnegative mixed derivatives. It satisfies the FKG lattice condition and hence is associated (Fortuin et al. 1971). For this continuous array, restrict to compact subintervals of \((0,\pi/2)\), apply the finite-grid FKG inequality to approximating densities, and pass to the limit. Bounded increasing functions follow by approximation and the usual monotone-class extension. The endpoint restrictions can then be removed.

For the plane-rotator assertion, write its components as \((\varepsilon\cos\gamma,\varepsilon'\sin\gamma)\), \(0\le\gamma\le\pi/2\). Given \(\gamma\), the signs are independent ferromagnetic Ising systems. The same mixed-derivative calculation with GKS proves association of \(\gamma\). Including amplitude factors, the first-component monomial mean is nonnegative and decreasing in \(\gamma\), while the second-component mean is nonnegative and increasing. Conditional independence gives the nonpositive covariance for unlike components. For like components, the conditional covariance is nonnegative by GKS and the covariance of conditional means is nonnegative by association.

In (10), a component-one monomial has conditional mean equal to its cosine amplitudes times a nonnegative linear combination of cosine-character means. It is nonnegative and decreasing in \(\alpha\). A component-four monomial has a nonnegative increasing conditional mean, by a fixed rotation of the right circle. Conditional independence gives the nonpositive covariance between these components. For like components, the conditional covariance is nonnegative by (9); the conditional means have the same monotonicity. Component symmetry supplies the remaining choices of components. ◻

Proof. We first control coefficients in the positive pairing expansion. This treats the odd case and several even cases. For the remaining case we must also control the covariance of conditional means.

The uniform sphere moment tensor vanishes in odd degree, and in degree \(2h\) it is \[\int_{S^3}q^{i_1}\cdots q^{i_{2h}}\,d\omega_3(q) =\frac{1}{4\cdot6\cdots(4+2h-2)} \sum_{\pi}\prod_{\{b,c\}\in\pi}\mathbf1_{\{i_b=i_c\}},\] where the sum runs over pairings of labeled factors. Expand each bond exponential in powers of the dot product and apply the tensor identity at each site. Following the contracted bond lines pairs the external factors, and every closed line contributes four. Thus, for arbitrary assigned external components, \[ \mathbb E_J\prod_{b=1}^{r+s}q_{x_b}^{i_b} =\sum_\pi d_\pi \prod_{\{b,c\}\in\pi}\mathbf1_{\{i_b=i_c\}}, \qquad d_\pi\ge0. \tag{12}\] The \(d_\pi\) do not depend on the component assignment. Repeated sites cause no difficulty because the factors remain labeled. Absolute convergence justifies regrouping the expansion. In particular the component two-point functions are nonnegative.

A pair is crossing if it joins \(\mathcal X\) to \(\mathcal Y\). When \(r,s\) are odd, every pairing crosses. Fix one crossing pair and assign component four to its endpoints and component one to all remaining endpoints. The resulting moment contains \(d_\pi\) as a nonnegative summand. Conditional on \(\alpha\), the remaining left-plane monomial has mean in \([0,1]\), including all amplitudes. Its removal bounds the moment by the chosen component-four pair mean. Hence \(d_\pi\le G\). Both separate monomial means vanish in odd total degree, so summing the pairing coefficients proves the odd case.

When \(r,s\) are even, every crossing pairing has at least two crossing pairs. Assign component four to one, component two to another, and component one to the remaining endpoints. In the left plane, Lemma 2 bounds the joint mean of the component-two pair and the component-one monomial by the product of their nonnegative means. The latter monomial mean and its amplitudes are at most one. The surviving component-two pair mean, including its amplitudes, is decreasing in \(\alpha\), because \[\mathbb E(\sin\theta_x\sin\theta_y\mid\alpha) =\tfrac12\mathbb E(\cos(\theta_x-\theta_y)\mid\alpha).\] This also holds when \(x=y\). The component-four pair mean is increasing in \(\alpha\). Association bounds their product expectation by the product of the full pair means, each at most \(G\). Therefore \[ d_\pi\le G^2 \quad\text{for every crossing pairing of two even lists}. \tag{13}\]

For two single-component even monomials, compare equal and different choices of their components. The products of their means agree by symmetry. Their joint moments differ by exactly the crossing contributions in (12). By Lemma 2, the two covariances have opposite weak signs, so each absolute covariance is at most this difference. Equation (13) proves the bound. The same reasoning applies to every pair of even sublists with the original value of \(G\). If \(P\) or \(Q\) has odd degree in some component, its separate mean is zero and every compatible joint pairing crosses. This also follows from (13).

It remains to consider monomials even in every component. We use single-component replacements, whose full covariances are already controlled, to bound both terms in the conditional covariance decomposition. Write \(P=P_LP_R\) by its two planes and put \[b_{P,L}=\mathbb E(P_L\mid\alpha),\qquad b_{P,R}=\mathbb E(P_R\mid\alpha).\] Let \(P_L^*\) replace all its components by component one and \(P_R^*\) replace all its components by component three. Denote their conditional means by \(B_{P,L},B_{P,R}\) and use the corresponding notation for \(Q\). Then \[|b_{P,L}|,|b_{P,R}|\le1,\qquad 0\le B_{P,L},B_{P,R}\le1,\] with \(B_{P,L}\) decreasing and \(B_{P,R}\) increasing in \(\alpha\).

Factor the common amplitude \(A_P(\alpha)\) from the left plane. Its Fourier expansion and the expansion after replacement are \[P_L=A_P(\alpha)\sum_k c_k\cos(k\cdot\theta),\qquad P_L^*=A_P(\alpha)\sum_k d_k\cos(k\cdot\theta),\qquad |c_k|\le d_k.\] Each sine or cosine supplies two exponential terms of modulus \(1/2\). Since the total number of sine factors is even, the original coefficients are real and invariant under \(k\mapsto-k\). Replacement changes their phases to positive coefficients. Collecting equal frequencies preserves their domination, including for repeated sites. The same argument applies in the right plane. By (9), \[ |\operatorname{Cov}(P_L,Q_L\mid\alpha)| \le\operatorname{Cov}(P_L^*,Q_L^*\mid\alpha), \tag{14}\] and likewise for \(R\).

Conditional independence gives \[\begin{align*} \operatorname{Cov}(P,Q\mid\alpha) ={}&\operatorname{Cov}(P_L,Q_L\mid\alpha) \mathbb E(P_RQ_R\mid\alpha)\\ &+b_{P,L}b_{Q,L}\operatorname{Cov}(P_R,Q_R\mid\alpha). \end{align*}\] The other factors have modulus at most one, so its averaged absolute value is at most \[\mathbb E\operatorname{Cov}(P_L^*,Q_L^*\mid\alpha) +\mathbb E\operatorname{Cov}(P_R^*,Q_R^*\mid\alpha).\] Each term is at most its full single-component covariance: the omitted covariance of conditional means is nonnegative by association. Both terms have already been bounded by \(C_{r,s}G^2\).

We still need the covariance of the conditional means. The Fourier comparison shows that \(B_{P,L}\pm b_{P,L}\) are nonnegative combinations of cosine-character means multiplied by their common amplitude. Both are decreasing; likewise \(B_{P,R}\pm b_{P,R}\) are increasing. For \(\alpha\le\widetilde\alpha\) this gives \[|b_{P,L}(\widetilde\alpha)-b_{P,L}(\alpha)| \le B_{P,L}(\alpha)-B_{P,L}(\widetilde\alpha),\] and the corresponding increasing bound for the right plane. The factors \(b_{P,R},b_{P,L}\) in the conditional mean need not be monotone. Their changes are nevertheless bounded by those of the increasing \(B_{P,R}\) and decreasing \(B_{P,L}\). To combine these two controls, put \[Y_P=b_{P,R}b_{P,L},\qquad D_P=B_{P,R}-B_{P,L}.\] Using the modulus bounds on the \(b\) functions yields \[|Y_P(\widetilde\alpha)-Y_P(\alpha)| \le D_P(\widetilde\alpha)-D_P(\alpha).\] Consequently \(D_P+Y_P\) and \(D_P-Y_P\) are increasing. Apply association to these functions and to \(D_Q\pm Y_Q\). Adding the two matching-sign inequalities and the two opposite-sign inequalities proves \[ |\operatorname{Cov}(Y_P,Y_Q)| \le\operatorname{Cov}(D_P,D_Q). \tag{15}\] On expanding the right side, the two same-plane terms are bounded by their full single-component covariances, as above. The two opposite-plane terms are exactly full single-component covariances by conditional independence; they enter with minus signs. Each is bounded in absolute value by the estimate already proved on even sublists. Empty sublists give constants and zero covariances. The total-covariance identity now proves the even case.

Finally, global spin inversion makes the joint moment zero when \(r+s\) is odd, and at least one separate mean is zero. This proves all cases of (11). ◻

Proof. The finite-graph estimate passes to the periodic plane limit for fixed monomials. Spatial translations are unitary in the OS form and commute with \(T_\beta\). Translated one-spin vectors have the same spectral support. Thus OS Cauchy–Schwarz gives \[0\le\mathbb E_{\mu_\beta} (q_{(0,x)}^1q_{(n,y)}^1)\le\tfrac14r_*^n\] for arbitrary spatial sites \(x,y\). The factor \(1/4\) is their one-site reflection norm squared.

A reflected polynomial transfer autocorrelation is a finite sum of monomial covariances at temporal separation at least \(n\). For an odd polynomial, (11) bounds it by \(C_P r_*^n\); for a centered even polynomial it bounds it by \(C_P r_*^{2n}\). The spectral measures are positive, so their supports lie in \([0,r_*]\) and \([0,r_*^2]\), respectively. Polynomials on a fixed finite spin product are dense in the \(L^2\) space of its marginal law. Reflection invariance gives \(\|[F]\|_{\mathrm{OS}}^2\le\mathbb E|F|^2\), so this approximation also holds in the OS seminorm. Centering and parity symmetrization preserve density. Bounded spectral projections extend the spectral bounds to the full subspaces. Since \([q_0^1]\) is centered, the full nonvacuum norm is exactly \(r_*\). Finally, the \(n\)th root of a moment of a positive spectral measure tends to the supremum of its support. This proves (16). ◻

Proof. The finite-graph estimate passes to the infinite-time cylinder by the row transfer representation. Component vectors \(q_x^i\Omega\) are centered, so their correlations across \(n\) rows are at most \(\tau^n\), suppressing \(\beta,\ell\). The even polynomial estimate therefore places the spectral support of every centered even polynomial vector in \([0,\tau^2]\). Polynomials are uniformly dense in continuous row functions, and the continuous positive \(\Omega\) has a positive minimum. Thus polynomials times \(\Omega\) are dense in row space, proving the even-sector norm bound.

Choose a real unit \(\tau\)-eigenfunction \(v\). Since \(\tau>\tau^2\), its eigenspace is odd under simultaneous inversion. The four component reflections commute with one another and with \(U\), so \(v\) can have definite parity under each; at least one is odd. Relabel it as component one. The modulus \(|v|\) is even. With \(b_0=\langle\Omega,|v|\rangle^2\), positivity of the kernel and the even-sector bound give \[\tau\le\langle|v|,U|v|\rangle \le b_0+(1-b_0)\tau^2, \qquad b_0\ge\frac{\tau}{1+\tau}.\] Take \(F=\operatorname{sgn}v\), equal to zero on its zero set. Its overlap with \(v\) is \(\langle v,F\Omega\rangle=\langle|v|,\Omega\rangle\). Keeping this contribution in the positive spectral expansion of its correlation proves (18). ◻

Proof. On a finite torus, write \[q_i=(\varepsilon_i u_i,\sqrt{1-u_i^2}\,l_i),\qquad \varepsilon_i\in\{-1,1\},\quad u_i\in[0,1],\quad l_i\in S^2.\] The prior is a product: \(\varepsilon_i\) is fair, \(l_i\) is uniform, and \(u_i\) has density proportional to \(\sqrt{1-u_i^2}\). Given \(u,l\), the signs are a ferromagnetic Ising model with coupling \(\beta u_i u_j\). In its Edwards–Sokal representation (Edwards and Sokal 1988), an edge between equal signs is open with probability \(1-e^{-2\beta u_i u_j}\), and an edge between different signs is closed. Given the edges, cluster signs are independent and fair. If no cluster joins the detector rows, flip all clusters touching the first row. This reverses \(F\) and leaves \(G\) unchanged. Its conditional product mean is therefore zero. The absolute detector correlation is bounded by the row-connection probability, and then by the sum of pair-connection probabilities.

For a pair \(x,y\) the joint spin–edge representation gives \[ \mathbb E(q_x^1q_y^1) =\mathbb E[u_xu_y\mathbf1_{\{x\leftrightarrow y\}}]. \tag{20}\] Condition on all edges, directions \(l\), and amplitudes other than \(u_i\). After summing the signs, each closed incident edge has factor \(e^{-J}\) and each open edge has factor \(e^J-e^{-J}\), where \(J=\beta u_i u_j\). The cluster multiplicity depends only on the fixed edges. Raising \(u_i\in[0,1/4]\) to \(u_i+1/4\in[1/4,1/2]\) increases every open factor. Closed-edge factors and perpendicular factors \[\exp\bigl(\beta\sqrt{1-u_i^2}\sqrt{1-u_j^2}\,l_i\cdot l_j\bigr)\] change by bounded ratios depending only on \(\beta\) and degree four. The amplitude prior is bounded above and below on these intervals. Comparing their integrals proves \[\mathbb P(u_i\ge1/4\mid\text{all other data})\ge c_\beta>0\] uniformly in the volume. Applying this twice by the tower property gives \(\mathbb E[u_xu_y\mid\text{edges},l]\ge c_\beta^2/16\). The connection event is edge-measurable, so (20) bounds its probability by \(16c_\beta^{-2}\mathbb E(q_x^1q_y^1)\). Summing proves the finite-torus estimate.

At fixed circumference the transfer representation converges to the cylinder for all bounded Borel row insertions: separate the ground projection, use their bounded multiplication norms, and let the excited trace tail tend to zero. The estimate therefore passes to the cylinder, including the sign detector. ◻

Source and applicability. Admission and coverage are (OpenAI 2026b, Proposition 7.1); calibration is its Proposition 6.2. The seed (21) is the concluding estimate in the proof of its Theorem 9.2: the reference depth and then the dyadic \(M_0\) are fixed before the final depth varies. The upper bound is its Proposition 10.2. Periodic-limit uniqueness and the all-temperature full gap are its Theorem 1.1 and Corollary 1.2. The sphere probability convention, single counting of bonds, even periods, and site-reflection one-step transfer coincide with ours. In particular the cited gaps concern the full local OS space, including rotation-invariant observables. ◻

Proof. At \(w=2t\), \(p(t,2t)\le\Delta(t)\). For widths \(z=t,2t\), advance the half-circumference estimate (29) to time \(w\) using (27) and \(E(z)\ge c\). This gives \(p(w,t)+p(w,2t)\le Ce^{-cw}\), after decreasing \(c\). The rectangle identity therefore gives \(p(t,2w)\le2p(t,w)+Ce^{-cw}\). Iteration over dyadic widths proves the result. ◻

Completion of Proposition 8. Apply (27) at time \(w\), with \(t_0\asymp w^{1/4}\) and widths \(W=w,2w\). The long-rectangle bound gives \(s(t_0,W)\le1\), and hence \[e^{-wE(W)}\le s(w,W)\le e^{-(w-t_0)E(W)}.\] Now \(E(W)\) is uniformly bounded and \(p(w,W)\) lies between \(s(w,W)/2\) and \(2s(w,W)\). Therefore \[-w^{-1}\log p(w,W) =E(W)+O_H(w^{-3/4}) =m(\beta)/a+O_H(w^{-1/4}).\] The positive sum \(\Delta(w)=2p(w,w)+p(w,2w)\) has the same logarithmic rate within this error. This proves (25). ◻

Proof. For \(n_{\mathrm p}=0\) the assertions are immediate. For positive particle number, the proof uses a common cone of componentwise nonnegative wavefunctions. We first show that the transfer kernel is strictly positive in this cone and that the contact Hamiltonian has a unique positive ground state. The commutation identity will then identify the transfer norm on that state.

Write the rows of \(Q\) as \[Q=\begin{pmatrix} c_0e^{i\phi}&s_0e^{i\theta}\\ -s_0e^{-i\theta}&c_0e^{-i\phi} \end{pmatrix}, \qquad c_0,s_0\ge0,\quad c_0^2+s_0^2=1.\] Change the sign of the color \((2,1)\). Every matrix-entry insertion now has a nonnegative radial amplitude times a single angular monomial in \((v,\phi,\theta)\). In a fixed-weight transfer kernel, the total angular charge vanishes. Separation of variables expresses the intervening heat operators, at each angular frequency, as radial heat operators with nonnegative killing potential. Their kernels are strictly positive in the radial interior. One may first impose Dirichlet boundaries on compact subintervals of \(0<c_0<1\) and then let these intervals increase. The two coordinate degeneracies have codimension two and are polar; the same separated heat kernel is consequently obtained for every angular frequency. This proves almost-everywhere strict positivity of the fixed-weight kernel of \(T_{n_{\mathrm p}}\) in the stated sign convention.

The same convention makes the contact Hamiltonian cooperative. If either binary index of a contacting pair agrees, the interaction is diagonal on the symmetric contact trace. If both indices differ, the left exchange changes the number of colors \((2,1)\) by one, and its off-diagonal entry is negative after the sign change. To use this observation without imposing artificial boundaries at coincidences, regard a wavefunction as a list of species components, each symmetric among particles of the same color. The diagonal operator in each component is a scalar Laplacian with scalar contact terms. Its heat kernel is positivity improving on the full periodic position torus.

Here is a direct justification that also supplies positivity of contact traces. Mollify each scalar delta interaction in its normal difference coordinate. For bridge times in \([1,2]\), the expected time-integral of these mollified interactions is bounded uniformly in the bridge endpoints and mollifier scale. Split the time interval in halves and use the free heat-kernel bounds together with the one-dimensional normal-coordinate bound \(C(1+t^{-1/2})\) near either endpoint. Feynman–Kac and Jensen’s inequality give a strictly positive lower bound for the mollified scalar heat kernel. The hyperplane trace estimate passes this domination to the contact operator: translation of a trace through normal distance \(\varepsilon\) changes its \(L^2\) norm by at most \(C\sqrt\varepsilon\|u\|_{H^1}\), and the associated resolvent forms are uniformly coercive at a sufficiently negative spectral parameter. Integration of the heat-kernel bound gives a positive lower bound for the scalar resolvent applied to any nonzero nonnegative component, including its contact trace. A nonnegative source supported only on contact hyperplanes also has nonnegative resolvent by the form maximum principle.

The absolute-value form inequality permits a ground vector to be chosen nonnegative. In its component equations the off-diagonal contacts give nonnegative sources. Moving a sufficiently large positive multiple of the component to the right-hand side produces a nonzero interior source whenever that component is nonzero; the preceding scalar resolvent argument then makes it positive, with positive contact trace. To see that positivity reaches every species component, write its four color counts as \(n_{11},n_{12},n_{21},n_{22}\). Their row sums and column sums are fixed by the two weights. Thus \(n_{11}\) determines all four counts and ranges over an interval of integers. The exchange \((11),(22)\leftrightarrow(12),(21)\) changes \(n_{11}\) by one and connects successive admissible counts. Positivity therefore propagates throughout the sector. Permutations and the positive multiplicities arising from identical-species symmetry do not affect this argument. If a real ground vector changed sign, the absolute-value inequality and the variational principle would make both its positive and negative parts nonnegative ground vectors with disjoint supports. This contradicts the strict positivity just proved. Ground-state simplicity follows.

It remains to connect this positive state to the Brownian transfer. The rotor representations are indexed by charge \(q\in\mathbb Z\) and spin \(j\in\frac12\mathbb Z_{\ge0}\), subject to \(2j\equiv q\pmod2\). Writing \(d=2j+1\), each representation contributes its \(d^2\) matrix elements and has energy \[E_{qj}=\frac{q^2+d^2-1}{2}.\] Multiplication by a matrix entry of \(G\) raises \(q\) by one and changes \(d\) by one in either allowed direction. In the variables \(x=(q+d)/2\), \(y=(q-d)/2\), it therefore raises exactly one variable by one. Let \[P(z)=\frac{4z^3-z}{3},\qquad J=P(x)+P(y).\] The finite-difference identity \(P(z+1)-P(z)=(2z+1)^2\) gives \[[E,[E,G_{ab}]]=[J,G_{ab}],\qquad [E,[E,\overline{G_{ab}}]]=-[J,\overline{G_{ab}}].\] In any ordering of insertion positions, derivatives of the rotor trace insert these commutators. Cyclicity makes the sum of the \(J\) commutators zero. Thus the sums of second derivatives in output and input positions agree away from coincidences.

At a coincidence of two output insertions, the derivative jump in either coordinate is \[[[E,G_{ab}],G_{cd}]=2G_{cb}G_{ad}.\] This follows from the Pauli identity, including the scalar generator; equivalently, for \(E=-\Delta/2\) the double commutator is the negative scalar product of the two gradients. The two kinetic derivatives hence give the contact term canceled by \(2cP^L\delta\) at \(c=2\). The conjugate computation handles two inputs, and at a mixed input-output coincidence the two delta terms cancel. We have proved the distributional intertwining identity for \(H_2\) and the kernel (37).

All differentiations can be justified before taking distributions. In each ordering, some heat interval has length at least \(W/(2n_{\mathrm p})\); multiplication by an entry shifts each angular label by only a bounded amount. The rotor series therefore converges with its one-sided derivatives; in particular, first derivatives are uniformly bounded on each closed ordering chamber. Continuity across its faces makes the kernel globally Sobolev \(H^1\) in either set of positions; intersections of several faces create no further boundary terms. The derivative bounds and the finite volume imply that \(T_{n_{\mathrm p}}:L^2\to H^1\) is bounded. If \(\mathfrak h_2\) is the form of \(H_2\), integration by parts in the distributional identity gives \[\mathfrak h_2(T_{n_{\mathrm p}}u,v) =\mathfrak h_2(u,T_{n_{\mathrm p}}v)\] first for smooth symmetric functions and then, by the trace bound and form-domain density, for all \(u,v\in H^1\). For \(u\in D(H_2)\) the right side equals \(\langle T_{n_{\mathrm p}}H_2u,v\rangle\), by self-adjointness of the transfer. Thus \(T_{n_{\mathrm p}}u\in D(H_2)\) and \(H_2T_{n_{\mathrm p}}u=T_{n_{\mathrm p}}H_2u\), which also proves resolvent commutation.

At zero twist, commutation maps the simple balanced ground space of \(H_2\) to itself. Its positive ground vector is therefore an eigenvector of \(T_{n_{\mathrm p}}\). Strict positivity of the latter kernel identifies that eigenvalue with its spectral radius in the sector, hence with its norm. The weight argument preceding the lemma gives the full norm.

It remains to extend commutation to the matching boundary twists used later. For real \(\zeta,\chi\), put \(g=\operatorname{diag}(e^{i\chi},e^{-i\chi})\) and \(\varepsilon_2=\left(\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\right)\). After the fixed change of exterior color basis \(\widetilde G=\varepsilon_2G\varepsilon_2^T\), use the rotor pullback \[(R_{\zeta,\chi}F)(v,Q)=F(v+\zeta,g^{-1}Qg^{-1}).\] Since \(\varepsilon_2g^{-1}=g\varepsilon_2\), this pullback transforms \(\widetilde G\) into \(e^{i\zeta}g\widetilde Gg\). Moving an output insertion through the seam therefore gives the particle color holonomy \(A=e^{i\zeta}g\otimes g\); an input insertion has the conjugate covariance. The transfer consequently maps sections with \(\psi(\ldots,X_i+W,\ldots)=A_i\psi(\ldots,X_i,\ldots)\) to sections with the same boundary condition. On rotor harmonics of charge \(q\) and spin \(j\), \(R_{\zeta,\chi}\) acts by \(e^{i\zeta q}\) and \(g_j^{-1}\) on both matrix indices. It commutes with \(E\) and \(J\), so cyclicity and the contact calculation remain valid. For these real twists the loop measure is positive and the boundary holonomy is unitary; the same Gram and weak-form arguments apply on the twisted Sobolev domain. Write \(\sigma_3=\operatorname{diag}(1,-1)\), with superscripts denoting its action on the indicated binary index. Multiplication by \[\prod_i\exp\!\left\{\frac{iX_i}{W} \bigl(\zeta I+\chi(\sigma_3^L+\sigma_3^R)_i\bigr)\right\}\] identifies this domain with the periodic one. In that fixed gauge the operator identities extend holomorphically to small complex twists; self-adjointness and positivity are used only for real twists. ◻

Proof. The kernel is positive, and its mass on \((-B,B)\) is bounded above by \(1-c/(1+B)\) uniformly in its center in the interval. A Neumann series therefore gives the inverse and the norm bound, as well as positivity of \(\rho_B\). On odd functions, restriction to \((0,B)\) replaces the kernel by \(K(x-y)-K(x+y)\), which is strictly positive for \(x,y>0\). Iteration gives a unique odd solution \(\eta_B\), positive for \(x>0\). Increasing \(B\) increases this solution on the old positive half-interval.

Define the even function \(\epsilon\) inside the interval by \(\epsilon(x)=\int_{|x|}^B\eta_B(y)\,dy\). Its boundary values vanish. Differentiating \(K*\epsilon_{\rm in}\) introduces no endpoint terms, so the equation for \(\eta_B\) shows that \((1-K*)\epsilon\) has the required logarithmic form up to an additive constant. At zero, changing the order of integration gives \[\epsilon(0)-(K*\epsilon_{\rm in})(0) =\int_0^B\left(1-2\int_0^xK(y)\,dy\right)\eta_B(x)\,dx =\int_0^B a(x)\eta_B(x)\,dx.\] This is (41). It proves strict monotonicity and continuity of \(b(B)\), and \(b(0)=\sqrt3\) by continuation. Extend \(\epsilon\) by the second equation in (40). To the right of \(B\), \[-\epsilon'(x)=\frac{2x}{x^2+1}+\frac{2x}{x^2+9} +(K*\eta_B)(x)>0,\] where positivity follows by pairing the odd integrand at \(y\) and \(-y\). Thus the extension is negative outside, as claimed. ◻

Proof. We construct \(U\) and \(V\) from the two transforms in (46), and then verify the convolution equation. Both transforms are analytic in \(\operatorname{Re}s>0\). Stirling’s formula gives, uniformly at infinity in the closed right half-plane, \[C(s)=1+O(s^{-1}),\qquad \frac{\sqrt\pi}{2sC(s)}= \frac{\sqrt\pi}{2s}+O(s^{-2}),\qquad \frac{\sqrt\pi}{2s}C(s)= \frac{\sqrt\pi}{2s}+O(s^{-2}).\] The Bromwich inverses therefore define locally bounded functions supported on \([0,\infty)\). More explicitly, subtract \(\sqrt\pi/(2(s+1))\) from either transform. The remainder is absolutely integrable on each vertical line in the right half-plane, so its inverse is continuous; closing the contour to the right shows that it vanishes for negative arguments. The subtracted transform has inverse \((\sqrt\pi/2)e^{-X}\) on \(X>0\). The \(O(s^{-2})\) remainder also shows that its inverse tends to zero as \(X\downarrow0\). This proves local boundedness, continuity for positive arguments, and the asserted right limits at zero. Both transforms commute with complex conjugation, so their inverses are real-valued.

To find the large-\(X\) behavior, expand at the origin. With \(\gamma_E\) denoting Euler’s constant, \[C(s)=\sqrt{s}\left[ 1+\frac{s}{2\pi} \bigl(\log s-\log(2\pi)-1+\gamma_E\bigr) +O\!\left(s^2(1+|\log s|^2)\right)\right].\] Consequently the transform of \(U\) is \[\frac{\sqrt\pi}{2}s^{-3/2} -\frac{\sqrt\pi}{4\pi}s^{-1/2} \bigl(\log s-\log(2\pi)-1+\gamma_E\bigr) +O\!\left(|s|^{1/2}(1+|\log s|^2)\right).\] The first term inverts to \(\sqrt X\). Differentiating \(\mathcal L(X^{\alpha-1})(s)=\Gamma(\alpha)s^{-\alpha}\) at \(\alpha=1/2\), and using \(\Gamma'(1/2)/\Gamma(1/2)=-\gamma_E-2\log2\), gives the second term of (47), including \(C'=\log(8\pi)+1\). For the remainder, deform the inversion contour to the rays \(\arg s=\pm3\pi/4\), joined by a small arc of radius \(1/X\). The principal branches are analytic in the intervening region, and Stirling’s formula is uniform in this sector. On the small part of the contour the displayed remainder contributes \(O(X^{-3/2}(1+\log^2X))\) after the substitution \(s=t/X\); on the remaining rays the exponential \(e^{sX}\) supplies the required decay. This justifies the termwise inversion and its error estimate. The same contour argument applied to the transform of \(V\), whose leading term at zero is \((\sqrt\pi/2)s^{-1/2}\), gives \(V(Y)=O(Y^{-1/2})\) for large \(Y\). Together with local boundedness, this is the stated bound on \(V\).

It remains to prove that the inverses satisfy the half-line equation. For real \(k\ne0\), the gamma-function identity \[\left|\Gamma\left(1+\frac{ik}{2\pi}\right)\right|^2 =\frac{|k|/2}{\sinh(|k|/2)}\] gives \[ C(ik)C(-ik)=1-e^{-|k|}. \tag{48}\] Use the Fourier convention \(\widehat f(k)=\int e^{-ikX}f(X)\,dX\). The polynomial growth just proved allows distributional Fourier transforms; the Laplace transforms give their boundary values away from \(k=0\). Since \(\widehat K_1(k)=e^{-|k|}\), (48) implies \[(1-e^{-|k|})\frac{\sqrt\pi}{2ikC(ik)} =-\frac{\sqrt\pi\,C(-ik)}{2(-ik)} \qquad(k\ne0).\] Thus the Fourier transform of \(U-K_1*U+\widetilde V\) is supported at zero, where \(\widetilde V(X)=V(-X)\) for \(X<0\) and \(\widetilde V(X)=0\) for \(X>0\). This assertion can equivalently be checked after inserting a positive Laplace regulator and then testing against functions supported away from zero. A distribution whose Fourier transform is supported at one point is a polynomial, so \(U-K_1*U+\widetilde V\) is a polynomial.

That polynomial vanishes. Indeed, write \(U(X)=\sqrt{X_+}+R(X)\), where \(X_+=\max(X,0)\). By (47), \(R\) is bounded and tends to zero at both infinities, and hence \(K_1*R\) also tends to zero there. The Poisson extension identity \[(K_1*\sqrt{\,\cdot\,_+})(X)=\operatorname{Re}\sqrt{X+i}\] shows that \(\sqrt{X_+}-K_1*\sqrt{\,\cdot\,_+}\) tends to zero at both infinities as well. Finally \(\widetilde V\) tends to zero. Therefore the polynomial tends to zero and is identically zero. Restricting this identity to \(X>0\) and \(X<0\) proves the two convolution identities. Their values at zero follow by continuity of \(K_1*U\) and the already established one-sided limits.

For positivity, \(U\) is positive for all sufficiently large \(X\) by (47), and its right limit at zero is positive. If its minimum on the positive half-line were negative, it would be attained at some \(X>0\). Averaging against \(K_1\), with the zero exterior value, would give a value strictly larger than that minimum, contradicting \(U=K_1*U\). Thus \(U\ge0\); the same averaging equation and positivity of the kernel make the inequality strict. The formula for \(V\) then gives \(V>0\). ◻

Proof. We first verify the cancellation that makes the approximants bounded. At the right endpoint set \(d=1-z\). The singular terms of the \(h\) row of (45) are \[\begin{aligned} F_{0,h}(z)&=\frac\pi2-\sqrt{2d}+O(d^{3/2}),\\ \delta F_{1,h}(z)&= -\frac{\delta}{2\pi\sqrt{2d}} \bigl(\log(d/\delta)+C'+i\pi\bigr)\\ &\quad+O\!\left(\delta\sqrt{|d|} (1+|\log\delta|+|\log|d||)\right). \end{aligned}\] Here the constant in the leading correction is exactly \(C'\) because \(2\pi C_0=\log\delta-\log(16\pi)-1\). The \(w\) row has the opposite square-root and inverse-square-root terms, in addition to its analytic terms. At the left endpoint the corresponding formulas are obtained by reflection and complex conjugation. They give precisely the signs used in the definitions of \(h_{\mathrm{app}}\) and \(w_{\mathrm{app}}\).

For clarity, the analytic expression for the two terms subtracted from \(U\) is \[ A(X)=\sqrt X+ \frac{\log X+C'+i\pi}{4\pi\sqrt X}, \qquad \operatorname{Im}X<0. \tag{49}\] Its real boundary value is the bracket defining \(D_0\) for \(X>0\), and is zero for \(X<0\). Since \(X=(1-z)/\delta\) lies in the lower half-plane when \(z\) lies in the upper half-plane, this is the appropriate continuation at the right edge; at the left edge we use its conjugate. Thus adding the \(D_0\) terms replaces the displayed singular terms by multiples of the locally bounded function \(U\). This proves the assertion about endpoint limits.

We estimate the residual before applying the interval comparison principle. Each function \(U(X_\pm(x))\) satisfies \((1-K_\delta*)U(X_\pm)=0\) for \(-1<x<1\), by the scaling of the half-line equation. After separating these exact terms from either approximant, write its remaining part as \(\operatorname{Re}(G_0+\delta G_1)\). Here \(G_0,G_1\) are obtained from the corresponding row of (45) by subtracting the analytic edge expressions in (49), with the signs just verified. The subtraction preserves the identities \[\operatorname{Im}G_0'=q,\qquad \operatorname{Im}G_1'=-\tfrac12\operatorname{Re}G_0'', \qquad q=\begin{cases}0&\text{for }h,\\1&\text{for }w.\end{cases}\] In particular \(\operatorname{Im}G_0'''=0\) on the interval. The removed terms leave endpoint remainders of order \(d^{3/2}\) in \(G_0\) and \(\sqrt d(1+|\log\delta|+|\log d|)\) in \(G_1\).

The remaining analytic functions are bounded at the endpoints and have sublinear growth at infinity, so convolution with \(K_\delta\) is their Poisson extension: \((K_\delta*\operatorname{Re}G)(x)=\operatorname{Re}G(x+i\delta)\). Put \(d_0=1-|x|\). If \(d_0\ge2\delta\), expand \(G_0\) to third order and \(G_1\) to first order on the vertical segment from \(x\) to \(x+i\delta\). The derivative identities cancel the term of order \(\delta^2\), and the cubic term from \(G_0\) has zero real part. The derivative bounds supplied by the endpoint remainders therefore give, for either approximant \(f_{\mathrm{app}}\), \[ \left|(1-K_\delta*)f_{\mathrm{app}}(x)-q\delta\right| \le C\left[ \delta^4d_0^{-5/2} +\delta^3(1+|\log\delta|+|\log d_0|)d_0^{-3/2} \right]. \tag{50}\] If \(d_0\le2\delta\), use the endpoint remainders directly at \(x\) and \(x+i\delta\). Their contributions are bounded by \(C\delta^{3/2}|\log\delta|\). The derivative of the nonanalytic \(O(d^{3/2})\) part of \(G_0\) tends to zero at the endpoint. Hence \(\operatorname{Im}G_0'=q\) identifies the imaginary part of the analytic linear coefficient there as \(q\), also at the left endpoint by reflection. The analytic terms therefore contribute \(q\delta+O(\delta^2(1+|\log\delta|))\). Hence the residual in this region is also at most \(C\delta^{3/2}|\log\delta|\).

On the exterior, the correction from the nearest endpoint vanishes. The argument of \(D_0\) at the opposite endpoint is at least \(2/\delta\), so (47) bounds that correction by \(C\delta^2(1+\log^2\delta)\). The uncorrected outer functions have exactly the step and zero exterior data, respectively. The step differs from the exterior data of \(h\) by an exponentially small quantity. Thus the exterior discrepancy of either approximant is bounded by \(e_\delta=C\delta^2(1+\log^2\delta)\).

Finally, the mass of the kernel that leaves the interval is \[\lambda_\delta(x) =\int_{\mathbb R\setminus(-1,1)}K_\delta(x-y)\,dy \ge c\frac{\delta}{d_0+\delta}.\] Dividing (50) by this lower bound, and using the bound in the region \(d_0\le2\delta\), gives \[\left|(1-K_\delta*)f_{\mathrm{app}}(x)-q\delta\right| \le C\delta^{3/2}|\log\delta|\,\lambda_\delta(x) \qquad(-1<x<1).\] For example, when \(d_0\ge2\delta\), the two ratios are bounded by \(C\delta^3d_0^{-3/2}\) and \(C\delta^2(1+|\log\delta|+|\log d_0|)d_0^{-1/2}\), respectively, and both have the asserted size. The nonnegative function \[e_\delta+M\delta^{3/2}|\log\delta|\, \mathbf 1_{(-1,1)}\] is consequently a comparison bound for both signs of the difference between the exact solution and its approximant, once \(M\) is large enough. Indeed the constant has zero Poisson difference, while the interval indicator has Poisson difference \(\lambda_\delta\) inside. The positivity of \(K_\delta\) and the strict loss of mass on the interval give the comparison principle. Since \(e_\delta=o(\delta^{3/2}|\log\delta|)\), this proves the claimed uniform estimate. ◻

Proof. For \(J=\frac12\), the local matrix is \(\mathcal L_{1/2}(x)=(x+i)-2iP\). Write its monodromy entries as \(A,B,C,D\). The relation with intertwiner \(u-v-2iP\) shows first that the \(B\) entries commute. It also gives the following coefficients when a diagonal entry is moved to the right past \(B(a)\): the terms that retain its spectral variable \(u\) have factors \[\frac{u-a+2i}{u-a}\quad\hbox{for }A(u), \qquad \frac{u-a-2i}{u-a}\quad\hbox{for }D(u).\] The terms that replace this variable by \(a\) have the opposite pole coefficients and contain \(B(u)A(a)\) or \(B(u)D(a)\). The all-up eigenvalues of \(A(u)\) and \(D(u)\) are \(F(u-i)\) and \(F(u+i)\).

For several \(B\) entries, first commute a selected \(B(a_r)\) past the other \(B\)’s. The coefficient of the term in which \(B(a_r)\) is replaced by \(B(u)\) is its pole coefficient times the keep factors for every other auxiliary variable, evaluated at \(a_r\). This also follows directly by induction and partial fractions. The two such coefficients, from \(A\) and \(D\) in the twisted trace, cancel precisely by the second equation of (57). The terms with no replacement give (58) for \(J=\frac12\).

For the higher auxiliary spins, multiply \[\mathcal L_{1/2}(x+2iJ)\mathcal L_J(x-i).\] Relative to the two total auxiliary spins \(J+\frac12\) and \(J-\frac12\), this product is triangular with diagonal blocks \[(x+i(2J-1))\mathcal L_{J+1/2}(x), \qquad (x+i(2J+1))\mathcal L_{J-1/2}(x-2i).\] To verify the blocks, let \(A\) and \(B\) be the two auxiliary angular momenta. The Pauli identity and \([A\cdot B,A]=iA\times B\) reduce the expansion to the two total-spin projections; the projection of \(A\) on these blocks is \(\pm(A+B)/(2J+1)\). The same calculation proves invariance of the plus block. Taking twisted traces yields a recurrence in \(J\). Substitution of (58) verifies the recurrence and its \(J=0,\frac12\) initial values, proving the formula for all \(J\).

It remains to impose the boundary condition. Move a momentum initially in the first slot once around the circle. The spin-\(\frac12\) trace is tested at \(k_j-i\), where the local factor at that slot is a permutation times \(-2i\). In each index the resulting transport is the twisted trace divided by \(-2i\); the joint scalar denominator from the other slots is \[\prod_{t\ne j}\bigl((k_j-k_t)^2+4\bigr).\] Together with the plane-wave phase this is exactly the first equation of (57). Braid consistency allows any momentum to be moved into the first slot, so every boundary condition follows. The contact conditions were already imposed by (56). Finally the coefficients are rational functions times entire plane factors, which proves the asserted meromorphic continuation. At a nonzero limit the contact and periodic boundary conditions persist, as does the eigenvalue equation. ◻

Proof. The proof has three parts. The scattering identities show that taking a pair residue creates no further bound cluster. A heat-kernel formula on separated contours then supplies the initial condition. Finally, lowering those contours produces (59), with its normalization determined by the particle occupying the first slot at each step.

Scattering of pairs and cancellation of higher residues. Let \(\Pi_{\varepsilon\eta}\), with \(\varepsilon,\eta\in\{+1,-1\}\), be the joint spectral projections of \(P^L,P^R\) on two slots. The interchange matrix has the decomposition \[\check R(x)= \frac{x-ic}{x+ic}\Pi_{++} +\frac{x+ic}{x-ic}\Pi_{--} -\Pi_{+-}-\Pi_{-+}.\] In particular, \[\mathop{\rm Res}_{x=ic}\check R(x)=2ic\,\Pi_{--}, \qquad \check R(0)=-I.\] Here \(\Pi_{--}\) is precisely the double-singlet projection. Taking this residue produces a low–high pair in the order specified above. Multiplication of two interchanges transports a pair past a single and multiplies its color vector by \[ S_{21}(x)= \frac{(x+ic/2)(x-3ic/2)} {(x-ic/2)(x+3ic/2)}. \tag{60}\] In this formula \(x\) is the center of the left block minus the momentum of the right block. If the single is on the left, the same formula holds with \(x\) equal to its momentum minus the pair center. Moving a pair past another pair gives \[ S_{22}(x)=\frac{x-2ic}{x+2ic}, \tag{61}\] where again \(x\) is the left center minus the right center. Both scalars have modulus one for real \(x\).

Here are explicit checks, including the partial exchanges that will occur during contour motion. In one binary index, for a pair \(AB\) followed by a single \(C\), set \[E(q)=s_{12}\otimes q_3,\qquad H(q)=q_1\otimes s_{23},\qquad F(q)=s_{13}\otimes q_2=E(q)+H(q).\] For an operator or vector \(V\) in one binary index, use \(V^{[2]}=V^L\otimes V^R\). Put \[D(x)=(x-ic/2)(x+3ic/2),\qquad T(x)=(x-ic/2)E+(x+ic/2)H.\] With \(x=u-k\), the coefficients of the three shuffles \(ABC,ACB,CAB\) are, respectively, \[E^{[2]},\qquad \frac{T(x)^{[2]}}{D(x)},\qquad S_{21}(x)H^{[2]}.\] These identities follow by applying the two permutation factors in \(\check R\) to the singlet. For a single initially on the left, interchange \(E\) and \(H\), and take \(x=k-u\). Thus neither partial nor complete exchanges have a pole at \(x=3ic/2\). At \(x=ic/2\), the last two residues in the pair-first order are \((ic/2)H^{[2]}\) and \(-(ic/2)H^{[2]}\). The elementary momentum then equals the lower member of the pair, so their plane phases agree and the residues cancel. In the single-first order the corresponding equality is with the upper member. This is also the cancellation expressed by \(\check R(0)=-I\).

For two pairs \(AB,CD\), both in low–high order, put \(x=u-v\). In one binary index set \[e=s_{12}s_{34},\qquad f=s_{14}s_{23},\qquad h=s_{13}s_{24}=e+f.\] The coefficients of the six allowed shuffles are \[\begin{array}{c|c} ABCD & e^{[2]}\\[2pt] ACBD & (xh+icf)^{[2]}/\{x(x+2ic)\}\\[2pt] ACDB & (x^2+c^2)f^{[2]}/\{x(x+2ic)\}\\[2pt] CABD & (x^2+c^2)f^{[2]}/\{x(x+2ic)\}\\[2pt] CADB & (xh-icf)^{[2]}/\{x(x+2ic)\}\\[2pt] CDAB & (x-2ic)e^{[2]}/(x+2ic). \end{array}\] This table can equally be obtained from \(P_{12}=-1\) on a singlet and \(1-P_{12}-P_{13}-P_{23}+P_{12}P_{23}+P_{23}P_{12}=0\) on three binary slots. It proves (61) and shows that the apparent poles at \(x=ic\) and \(x=2ic\) are absent. At \(x=0\), the four middle residues are, in their listed order, \[\frac{ic}{2}f^{[2]},\quad -\frac{ic}{2}f^{[2]},\quad -\frac{ic}{2}f^{[2]},\quad \frac{ic}{2}f^{[2]}.\] The phases agree in the pairs \(ACBD,ACDB\) and \(CABD,CADB\), because \(B\) and \(D\) have the same momentum. The cancellation therefore holds both in the whole wave and in either part in which the first slot is required to contain the low member of a specified pair: the part starting with \(A\) consists of the first three shuffles, and the part starting with \(C\) of the last three.

The heat kernel on separated contours. Let \(E_X^{n_p,0}(k)\) be the elementary-wave matrix, and choose horizontal contours \(\Gamma_j=\mathbb R+i\alpha_j\), where \[\alpha_{n_p}=0,\qquad \alpha_j-\alpha_{j+1}>c.\] Consider \[ \mathcal K_s(X,Y)= \int_{\Gamma_1\times\cdots\times\Gamma_{n_p}} E_X^{n_p,0}(k) e^{-i\sum_j k_jY_j-s\sum_j k_j^2} \prod_j\frac{dk_j}{2\pi}. \tag{62}\] Gaussian decay makes this integral and its spatial derivatives convergent on compact sets. Each plane wave solves the free heat equation in the chamber, and the interchange relation imposes the contact conditions on each face.

To verify the initial condition, one needs the support of the Fourier transforms of the rational coefficients. For \(\operatorname{Im}x>c\), the preceding projector decomposition gives the concrete formula \[\check R(x)=P^LP^R+ 2c\int_0^\infty e^{ixt} \bigl(e^{ct}\Pi_{--}-e^{-ct}\Pi_{++}\bigr)\,dt.\] Accordingly, an inversion of labels \(i<j\) contributes a nonnegative variable \(t_{ij}\), possibly an atom at zero, and a phase \(e^{i(k_i-k_j)t_{ij}}\). If a permutation puts label \(i\) at position \(p(i)\), its zero-time contribution can be supported only at \[ Y_i=X_{p(i)}+ \sum_{\substack{j>i\\p(j)<p(i)}}t_{ij} -\sum_{\substack{h<i\\p(h)>p(i)}}t_{hi}. \tag{63}\] For a nonidentity permutation, let \(r\) be the first nonfixed position, and let \(j>r\) be the label occupying it. Label \(r\) occurs at some position \(p(r)>r\). No smaller label contributes a negative shift to label \(r\); no larger label contributes a positive shift to label \(j\). Thus (63) would imply \[Y_r\geq X_{p(r)}>X_r\geq Y_j,\] which is incompatible with \(Y\in\mathcal C_{n_p}\). Only the identity permutation remains, and its coefficient is the identity color operator. It supplies the delta initial condition.

The support argument also supplies the estimates needed to justify this limiting calculation and heat-kernel uniqueness. Let \(d\) be the shift vector on the right side of (63), so that \(z_i=X_{p(i)}+d_i\). The shifts belong to the cone generated by \(e_i-e_j\), \(i<j\). This cone is pointed, and more explicitly \[\sum_{i<j}t_{ij} \leq \sum_{i<j}(j-i)t_{ij} =-\sum_i i\,d_i \leq C_{n_p}|d|.\] Furthermore, \[\sum_i X_{p(i)}d_i =\sum_{\substack{i<j\\p(j)<p(i)}} t_{ij}(X_{p(i)}-X_{p(j)})\geq0, \qquad |z|^2\geq |X|^2+|d|^2.\] The measures in the transform formula have at most exponential growth in \(\sum t_{ij}\). Gaussian convolution therefore dominates them and gives spatial decay for compactly supported initial data. It also justifies convergence to the stated initial condition, first for data supported strictly inside the chamber and then by density. The contact form is lower bounded by the hyperplane trace inequality, so uniqueness of its \(L^2\) heat evolution identifies \(\mathcal K_s\) with \(e^{-sH_c}\).

Lowering the contours and reconstructing the adjoint wave. Process the momenta from last to first. The induction assertion can be stated explicitly. After the last \(n_p-h\) slots have been processed, sum over \(\ell'+2m'=n_p-h\), with weight \((2c)^{m'}/(\ell'!m'!)\). The first \(h\) elementary momenta \(\xi_1,\ldots,\xi_h\) remain on their original high contours; the suffix momenta \(k\) and pair centers \(u\) are real. The wave factor in the integrand is \[E_X^{h+\ell',m'}(\xi,k,u) \left[I_{\mathcal V_h}\otimes E_{(Y_{h+1},\ldots,Y_{n_p})}^{\ell'm'}(k,u)^\dagger\right] e^{-i\sum_{a=1}^h\xi_aY_a}.\] Its Gaussian energy is \(\sum\xi_a^2+\sum k_a^2+2\sum u_a^2-m'c^2/2\). Here the \(X\)-wave uses the first \(h\) elementary slots followed by the suffix blocks as its initial order; its finite meromorphic formula also defines it for the complex variables \(\xi\). Thus \(h=n_p\) is (62), whereas \(h=0\) is precisely (59).

To pass from \(h\) to \(h-1\), decompose the new suffix wave according to its first occupied slot. For each elementary carrier or labeled pair \(a\), let \(E_Y^{[a]}\) be the partial sum in which that slot contains, respectively, the elementary momentum or the pair’s low member. These possibilities are disjoint and exhaust the allowed shuffles, so \(E_Y=\sum_a E_Y^{[a]}\). The ordinary contour integral will supply the elementary-first partial adjoints, and its residues will supply the pair-first partial adjoints. When block parameters \(z\) leave the real contours, an adjoint wave is continued as the meromorphic function \(E_Y(\overline z)^\dagger\), which agrees with \(E_Y(z)^\dagger\) for real \(z\). The same convention applies to each partial adjoint.

Let \(k=\xi_h\) be the next momentum. The integral left after lowering its contour to the real line provides the adjoint-wave terms whose first suffix slot is an elementary momentum. To see this with a fixed elementary carrier, put that carrier first among the suffix blocks. The unitary elementary and whole-block interchanges move the corresponding coefficient from the \(X\)-wave to the \(Y\)-wave. Symmetrizing the elementary variables then supplies all elementary choices for the first slot.

The residues occur at \(k=p+ic\), where \(p\) is an elementary momentum in the old suffix. Put \(p\) first in that suffix before computing the residue. A downward contour shift contributes \(-i\) times the residue, so the residue \(2ic\,\Pi_{--}\) contributes the factor \(2c\) and inserts a normalized double singlet. The momenta at \(X\) are \(p,k\), in low–high order, with center \(u=p+ic/2\). Lower \(u\) from this contour to the real line. On the \(Y\) side the new first slot uses the adjointed phase of the low member, namely the phase with the high momentum \(k\). The other member is the old suffix variable \(p\). The old suffix exchanges are continued with this \(p\), and hence give exactly the part of \(E_Y^\dagger\) with that pair’s low member first.

For completeness, the pole checks justifying both contour motions are as follows. When \(k\) descends past an existing pair, the three-slot formulas have no pole at center difference \(3ic/2\); the possible pole at \(ic/2\) cancels between the partial and complete exchanges whose phases agree. There is therefore no three-particle residue. When the newly formed center \(u\) is lowered, the same cancellation applies to the \(X\)-wave. In the restricted adjoint \(Y\)-expression the low member of the new pair has already been put first. The crossing that remains uses only \(p=u-ic/2\). Its imaginary part moves from \(0\) to \(-c/2\), and no elementary interchange pole at difference \(\pm ic\) is crossed. This includes an elementary momentum on the old real contour, by continuation with an indentation if necessary. Explicitly, the adjoint of the pair-low-first three-slot sum has denominator \((x+ic/2)(x-3ic/2)\), with \(x=u-k_{\rm single}\), which has no zero in \(0\leq\operatorname{Im}x\leq c/2\). For two pairs, the four-slot table proves regularity at coincident centers even in the restricted first-slot expression. The only other possible center pole is at \(-2ic\) for the \(X\)-coefficients and at \(2ic\) for the continued adjoint coefficients, both outside the strips being traversed.

These checks remain valid with other variables inserted. At a generic point of a candidate pole hyperplane, use the braid and whole-block identities to bring the two relevant blocks together. For fixed occupied slots, inserting the other momenta multiplies the two cancelling residues by the same nonsingular interchange operators. If the first slot is restricted, put its block first before this operation. The earlier, unprocessed contours stay separated from all these motions. Finally, write the finite sums over a common rational denominator. Removal of the poles at generic points of every denominator hyperplane removes the corresponding factors, so the removals extend across their intersections. Distinct real parts or indented contours may therefore be used for the residue computations and subsequently removed. On bounded \(X,Y\), small contour circles about removable singularities give uniform bounds for the finite wave sums. Together with the Gaussian, these bounds justify the integrations and deformations just made.

It remains to give every partial adjoint the same weight. Changing the initial block order multiplies both waves by the same unitary block-interchange operator, so the product \(E_X(E_Y^{[a]})^\dagger\) is independent of that choice. We may therefore put each first carrier in front before symmetrizing the labels. For a final sector \((\ell',m')\), the elementary-first contribution comes from \((\ell'-1,m')\) in the old suffix and has weight \[\frac{(2c)^{m'}}{(\ell'-1)!\,m'!} =\ell'\frac{(2c)^{m'}}{\ell'!\,m'!}.\] It accounts for the \(\ell'\) choices of elementary first carrier. The pair-first contribution comes from \((\ell'+1,m'-1)\). There are \(\ell'+1\) elementary choices for \(p\), and the residue supplies \(2c\); its weight is therefore \[2c(\ell'+1) \frac{(2c)^{m'-1}}{(\ell'+1)!\,(m'-1)!} =m'\frac{(2c)^{m'}}{\ell'!\,m'!}.\] It accounts for the \(m'\) choices of pair first carrier. After symmetrization, each of the \(\ell'+m'\) partial adjoints has weight \((2c)^{m'}/(\ell'!m'!)\). Their sum is the full adjoint wave. This closes the induction and proves (59).

Define the transform explicitly by \[(\mathcal F_c f)_{\ell'm'}(k,u)= \left(\frac{(2c)^{m'}}{\ell'!\,m'!}\right)^{1/2} \int_{\mathcal C_{n_p}}E_Y^{\ell'm'}(k,u)^\dagger f(Y)\,dY,\] initially for compactly supported \(f\). Its target is the direct sum of the \(L^2\) spaces with the product measures \(dk\,du/(2\pi)\) in (59) and color space \(\mathcal V_{\ell'}\). Writing \[\lambda_{\ell'm'}(k,u)= \sum k_a^2+2\sum u_a^2+\frac{(M-m')c^2}{2}\geq0,\] the quadratic identity from (59) is \[\langle f,e^{-s(H_c+Mc^2/2)}f\rangle =\sum_{\ell'+2m'=n_p}\int e^{-s\lambda_{\ell'm'}}|(\mathcal F_cf)_{\ell'm'}|^2.\] As \(s\downarrow0\), strong continuity on the left and monotone convergence on the right give \(\|\mathcal F_cf\|=\|f\|\). The transform thus extends isometrically to the full Hilbert space. The same identity then implies \(\|e^{-s(H_c+Mc^2/2)}\|\leq1\), and hence \(H_c+Mc^2/2\geq0\). Taking the increasing limit of \((1-e^{-s\lambda})/s\) gives, by the spectral theorem, the stated quadratic-form identity and its domain. The reconstruction \(\mathcal F_c^*\mathcal F_c=I\) is all that will be needed; surjectivity onto the direct sum is unnecessary. ◻

Proof. Divide all momenta by \(c\). A shuffle’s plane phase has modulus \[\exp\left\{-\frac c2 \sum_{\text{pairs }a} (X_{\text{high slot of }a} -X_{\text{low slot of }a})\right\}.\] The sum of the pair spans is at least the sum of the component spans. Rational interchange coefficients are uniformly bounded away from their pole hyperplanes. Near real centers, their only apparent singularities are coincident pair centers. The four-slot cancellations in the preceding proof are internal to the present groups: switching the equal high members preserves the slots and carriers of each overlapping component. More explicitly, the cancelling shuffles are \(ACBD,ACDB\) and \(CABD,CADB\). Within either pair of shuffles, the union of the spans belonging to \(AB\) and \(CD\) is the same interval, from their earliest low member to their latest high member. Inserting further slots does not change that equality. The connected components obtained after adjoining all other pair spans, and their sets of carriers and included singles, are therefore unchanged. Thus every group sum is holomorphic through these real hyperplanes.

Here is a uniform Cauchy estimate for this holomorphic sum. Choose a sufficiently small number depending only on \(n_p\). Among finitely many successively much smaller scales, one can choose a scale with no distance between real divided momenta in the intervening annulus: there are only finitely many pairwise distances. Cluster the arguments at the smaller scale. Arguments in different clusters are then separated by much more than the chosen scale. Within each cluster, put the variables on small circles about their original real values, using distinct radii separated by fixed multiples of that scale. All radii can be kept below the initially chosen small number. On the product of these circles, equal-center denominators are bounded away from zero by a positive \(n_p\)-dependent constant; denominators with nonzero imaginary offsets are also separated from zero. Every rational coefficient is consequently bounded there by a constant depending only on \(n_p\). Cauchy’s formula recovers its holomorphic group sum at the original real arguments.

To control the plane phases on those circles, factor from each component the common phase evaluated at its leftmost position. The carrier set of that component is fixed throughout the group, so the factored phase is the same in every term and has modulus one at real momenta. Perturbing the remaining divided momenta by the small circle radii increases their modulus by at most \(\exp\{\varepsilon C_{n_p}c\sum_{\mathcal I} \operatorname{span}(\mathcal I)\}\). Choose the initial upper bound on the radii small enough that this uses less than half the original pair-gap decay. The resulting Cauchy estimate proves (64). ◻

Proof. The form localization identity gives \[\sum_\alpha \left\langle\vartheta_\alpha\psi_j, (H_{c_j}+Mc_j^2/2)\vartheta_\alpha\psi_j\right\rangle = \left\langle\psi_j, (H_{c_j}+Mc_j^2/2)\psi_j\right\rangle +\sum_\alpha\|\,|\nabla\vartheta_\alpha|\psi_j\|^2.\] The error is bounded independently of \(j\), and every localized term is nonnegative by Lemma 17. It suffices, therefore, to prove the assertion for one localized function on the line, with bounded norm and bounded shifted form energy. Fix a bounded chamber box containing the supports of all these localized functions. In what follows, restrict the output of every reconstruction kernel to this box. This is an \(L^2\) contraction and changes none of the localized functions; it is needed because a momentum-truncated reconstruction need not itself have compact support.

Apply the isometric transform of Lemma 17. Restrict all elementary momenta and pair centers to a fixed bounded set, say absolute value at most \(R\). The form identity bounds the squared norm discarded by this restriction by \(C'/R^2\). Its reconstruction has at most this norm because \(\mathcal F_c^*\) is a contraction. Likewise, all sectors with fewer than \(M\) pairs have \(\lambda_{\ell'm'}\geq c^2/2\); their total squared transform norm is at most \(2C'/c^2\). Thus only the sector with \(M\) pairs and \(\ell\) singles can survive, and its momenta may first be kept in a fixed bounded set.

For that sector the reconstruction kernel contains the factor \((2c)^{M/2}/\sqrt{\ell!\,M!}\). Consider a group with overlapping pair spans or with a single inside a pair span. If a component contains \(p\) pairs and \(s_0\) singles, it has \(2p+s_0\) slots and \(2p+s_0-1\) internal gaps. For a component consisting of one adjacent pair this is one gap. For an overlapping component or one containing an inserted single it is at least \(p+1\) gaps. Squaring (64) and integrating the ordered coordinates therefore gives one factor \(O(c^{-1})\) for every such gap. Over all components there are at least \(M+1\) decaying gaps in these groups. The squared kernel normalization is \(O(c^M)\), so their squared Hilbert–Schmidt norms on the momentum cutoff are \(O(c^{-1})\). The remaining component positions range over a fixed bounded set. The bounded momentum volume and the Cauchy–Schwarz inequality now show that all these groups have vanishing reconstructed \(L^2\) norm, uniformly for bounded transform coefficients.

Every surviving shuffle is a permutation of whole adjacent blocks. In coordinates for its adjacent-pair assignment, its normal plane factor is exactly \(\exp(-\sum z_a/2)\) and its color factor is the product of double singlets in (65). The coefficients between whole blocks are the explicit scalars (60)–(61), whose limits on a fixed momentum cutoff exist as \(c\to\infty\). The rescaled Jacobian cancels the factor \(c^{M/2}\) in the kernel normalization. The fixed center-coordinate volume, the finite momentum cutoff, and the exponential normal factor give an integrable dominating square. The reconstruction kernels for this assignment consequently converge in Hilbert–Schmidt norm on every retained chart. On a chart belonging to another assignment, their contribution tends to zero: bounded rescaled gaps for both assignments would force two distinct block centers to approach each other, whereas the retained chart separates those centers by \(\delta\). The exponential gap bound controls the remaining contribution.

This kernel convergence implies compactness even though the transform coefficients depend on \(j\). On the fixed cutoff their \(L^2\) norms are bounded; extract a weakly convergent subsequence. The limiting Hilbert–Schmidt operator is compact, and convergence to it in operator norm then gives strong convergence of the reconstructed functions. The limit has the asserted normal and color factors.

We verify that this argument retains the whole norm. Restrict first to centers separated by at least \(\delta>0\) and to \(0<z_a<Z\). Different adjacent-pair assignments then occupy disjoint configuration regions for all sufficiently large \(c\). The retained region is the union over those assignments; each whole-block kernel is estimated in the coordinates of its own assignment. For the bounded-momentum kernels, the squared norm excluded by center distances below \(\delta\) is uniformly small as \(\delta\downarrow0\): the kernels are uniformly bounded in the center coordinates, and these collision neighborhoods have vanishing volume. The norm excluded by \(z_a>Z\) tends uniformly to zero as \(Z\to\infty\), by the exponential normal bound. The groups already discarded cover the overlaps and inserted singles, and have vanishing norm. Thus neither intersections of pair coincidence manifolds nor their complement retain additional norm.

Finally remove the momentum cutoff using the uniform \(C'/R^2\) bound, and take a diagonal subsequence over the finitely many seam charts and pair assignments and a countable exhaustion of the separation and gap cutoffs. The exact identity \(\sum_\alpha\vartheta_\alpha^2=1\) then gives the asserted total norm. Applying the same cutoffs to two sequences, strong convergence on the retained charts gives convergence of their inner products; Cauchy–Schwarz controls all discarded parts. ◻

Proof. Trial states consisting of separated, exponentially bound pairs and smooth center wavefunctions give a circle ground-state energy at most \(-Mc^2/2+O(1)\). One can cut off the pair gaps inside fixed disjoint intervals: the cutoff error is exponentially small in \(c\), and the center kinetic energy is bounded. The true normalized ground state therefore satisfies Lemma 19. It is the simple ground state in this sector, is nonnegative in the static-sign convention, and its limits retain total norm \(1\). Their normal factors are (65); in a sector balanced in each binary index, the remaining single, if present, has the majority color in each index. A nonzero nonnegative limit consequently has strictly positive inner product with the strictly positive limiting center coefficient of the eigenbranch.

If the two eigenstates were different for an unbounded sequence of couplings, self-adjointness and the simplicity of the ground state would make their inner products zero on that sequence. Pass to a simultaneous compactness subsequence. The inner-product convergence just proved gives a strictly positive limit, a contradiction. This identifies the strong-coupling branch from its wavefunction and energy, without assuming a string description of the ground state on a finite circle. ◻

Proof. Move the increasing pair terms in the first equation to the left. For fixed \(v\), these equations are the gradient of a strictly convex function: the quadratic part has Hessian \(2\mathcal W I\), the pair interactions give a nonnegative weighted graph Laplacian, and the term involving \(\varphi\) has uniformly bounded diagonal curvature. The Hessian is therefore strictly positive. Coercivity gives existence and uniqueness. The increasing right-hand sides \(\pi(2j-M-1)\) imply the ordering of the \(u_j\); this follows also by interchanging two variables in the convex minimization problem. Reflection reverses the index order and changes every sign, so uniqueness gives \(u_j=-u_{M+1-j}\) when \(v=0\). Oddness of \(\varphi\) then proves the single equation.

The displayed formulas for \(Z_c\) and \(d\) follow by division and differentiation of (66). Since \(K\geq0\), its only possibly negative term is \(O(\mathcal W^{-1})\), giving the lower bound for \(d\). The total quantile interval has length \(M/\mathcal W=O(b^2)\), which gives the first bound for \(B_c\). The derivatives of each Cauchy kernel are bounded, and \(M/\mathcal W=O(b^2)\), proving the derivative bounds. The quantile increments are \(1/\mathcal W\), so the upper bound for \(d\) gives the stated separation.

For a smooth function of \(u\), summation over the centers is midpoint summation after the change of variable \(s=Z_c(u)\). The inverse change of variable and its fixed derivatives have polynomial bounds because \(d\) is bounded below. Applying this rule to \(K(x-u)\), uniformly in \(x\), gives \[\frac1{\mathcal W}\sum_jK(x-u_j) =\int_{-B_c}^{B_c}K(x-y)d(y)\,dy +O\!\left(\frac{\operatorname{poly}(b)}{\mathcal W^2}\right).\] The inverse of \(1-K*\) on this interval has norm \(O(1+B_c)\) by Lemma 12. Solving the resulting equation for \(d\) proves (67). To bound its middle term, use \(S*(\delta_0-K)=K_1-K_3\) and write \[(1-K*)_{B_c}^{-1}(K_1-K_3) =S_{\mathrm{in}} -(1-K*)_{B_c}^{-1}(K*S_{\mathrm{out}}).\] The inverse costs a polynomial in \(1+B_c\), whereas \(S_{\mathrm{out}}\) has an exponential tail. This gives uniform supremum and \(L^1\) bounds; bounded values of \(B_c\) are covered by the same inverse estimate.

Integrating (67) shows \[\int\rho_{B_c}\leq b^2/2+O(b).\] The total-density asymptotic in (43), together with its relation between \(b\) and \(B_*\), yields \(B_c\leq B_*+O(1)\). At \(\mathcal W=W\), the two-sided particle band yields the reverse estimate. The constants can be enlarged to include bounded \(B_c\).

It remains to check the Jacobian when the single is present. The pair block is a diagonally dominant matrix with nonpositive off-diagonal entries, and its inverse has absolute row sums \(O(\mathcal W^{-1})\). At the symmetric solution the single diagonal entry is \[ \mathcal W-\sum_j\varphi'(-u_j) =2\pi\mathcal W\, S*(\rho_{\mathrm{ext}}\mathbf1_{|x|>B_c})(0)+O(1), \qquad \rho_{\mathrm{ext}}=1/\pi+K*\rho_{B_c}. \tag{68}\] Here \(\rho_{\mathrm{ext}}=\rho_{B_c}\) inside the interval. To see the identity, use \(\varphi'=2\pi(K_1-K_3)\), apply midpoint summation and (67), and then convolve \[\rho_{B_c}-K*\rho_{B_c} =1/\pi-\rho_{\mathrm{ext}}\mathbf1_{|x|>B_c}\] with \(S\). The impurity correction in (67) has total mass \(O(\mathcal W^{-1})\), so its contribution is \(O(1)\).

The edge lower bound in Proposition 13 and positivity of \(S\) bound the main term in (68) below by \[c\mathcal W\sqrt{1+B_c}\,e^{-\pi B_c/2}.\] The same conclusion for bounded \(B_c\) follows directly from positivity. The absolute sum of the cross entries is \[\sum_j|\varphi'(u_j)|=O((1+b)\mathcal W).\] For example, sum a smooth integrable envelope of \(|\varphi'|\) by the counting rule and use \(\rho_{B_c}\leq C(1+B_c)\leq C(1+b)\). Combining this with the pair inverse bound shows that the correction in the single Schur complement is \(O(1+b)\). Finally (43) and the circumference assumption give \[W\sqrt{1+B_*}\,e^{-\pi B_*/2} \asymp b(\log b)^D.\] This dominates the Schur correction and the bounded remainder by a factor tending to infinity. The bound improves for \(\mathcal W\geq W\), since \(B_c\leq B_*+O(1)\). The Schur complement is therefore invertible, and block inversion gives the claimed polynomial bound for the full inverse. ◻

Proof. At the upper member of a formal pair, the external factors in the first equation of (57) contributed by another pair, at center difference \(x\), multiply to \[\frac{x(x+4i)}{x^2+4}.\] Its log modulus is \(A(x)\). The Fourier identity \[\widehat A(k)=-\pi\frac{(1-e^{-2|k|})^2}{|k|}\] implies \[\kappa=R*(\rho_{\mathrm{ext}}\mathbf1_{|x|>B_c}), \qquad \widehat R(k)=\pi\frac{\tanh|k|}{|k|}.\] Indeed \(\widehat R(1-\widehat K)=-\widehat A\) and \(\int R=\pi\). The partial fraction expansion of \(\tanh k/k\), obtained from the product for \(\cosh k\), gives, for \(x\ne0\), \[R(x)=2\sum_{m\geq0} \frac{e^{-(2m+1)\pi|x|/2}}{2m+1}.\] Thus \(R\) is positive and \(R(x)\geq2e^{-\pi|x|/2}\). The edge lower bound for \(\rho_{B_c}\) also bounds \(\rho_{\mathrm{ext}}\) below by a constant times \(\sqrt{1+B_c}\) in a fixed interval just outside each endpoint. Convolution over these intervals proves the first inequality in (69). Its final comparison follows from the scale estimate at the end of Lemma 21; for larger \(\mathcal W\) the left side only gains the needed margin.

For the sum, apply midpoint summation in the coordinate \(s=Z_c(u)\). Away from the cell containing \(u_j\) and its two neighbors, monotone or bounded-variation summation for the logarithmic singularity costs \(O(\log\mathcal W)\). The spacings from Lemma 21 show that the log integral over each of the excluded quantile cells is at most \(C\log\mathcal W/\mathcal W\). This gives the same total cost for those cells. Finally replacing \(d\) by \(\rho_{B_c}\) uses (67): the inverse function there is bounded, \(A\) is integrable, and the remaining error is polynomial in \(b\) times \(\mathcal W^{-1}\). ◻

Proof. We first rewrite the equations at the formal pair spacings without dividing by a vanishing internal factor. The resulting system has a polynomially bounded inverse linearization, while its error is smaller than every inverse power of \(\mathcal W\); this will give the true roots by contraction.

The \(j\)th auxiliary equation can be written \[\frac{X_j}{2i+X_j}\frac{Y_j-2i}{Y_j}=A_j,\] where \(A_j\) is analytic and contains only external factors and the twist. Set \(X_j=D_jY_j\) and impose \[D_j=A_j\frac{2i+X_j}{Y_j-2i}.\] At the formal solution \(|A_j|=1\) and \(D_j=-A_j\), so \(D_j\) stays away from zero. Denote by \(F_j\) the product in the upper-member momentum equation after removing its internal pair factors. Solving this equation for \(Y_j\) gives the regular relation \[ Y_j=(D_j-1) \frac{e^{ik_{j+}\mathcal W}(2i+X_j)^2} {F_j(4i+X_j-Y_j)D_j^2}. \tag{71}\] In addition, retain the product of the two momentum equations, replacing the internal auxiliary ratio by \(A_j\), and retain the single equation when present. The logarithms of these last equations are taken near (66). They are analytic there. Distinct pair centers are separated; the single need not be separated from a center, because its potentially dangerous factors have nonzero imaginary shifts.

We give the size estimates for this regular system. On a neighborhood of radius \(\mathcal W^{-C}\), with a fixed sufficiently large \(C\), the logarithmic derivatives through two orders of the nonzero external factors, and the row sums of the logarithmic derivatives of their products, are bounded by fixed powers of \(\mathcal W\). This follows from the center spacing bound and the polynomial number of factors. Increasing \(C\) therefore makes the variation of each product logarithm arbitrarily small. We do not require a polynomial bound for the raw product \(F_j\) or its derivatives. For \(A_j\), whose formal modulus is one, these logarithmic bounds give polynomial bounds for the ordinary derivatives. For the upper-member equation the bounded quantity is instead the normalized ratio \(e^{ik_{j+}\mathcal W}/F_j\). Indeed, at the formal centers the logarithm of the modulus of this ratio is \[\log\left|\frac{e^{ik_{j+}\mathcal W}}{F_j}\right| =-\mathcal W-\sum_{t\ne j}A(u_j-u_t)+O(\log\mathcal W) =-\mathcal W\kappa(u_j)+O(\log\mathcal W).\] The possible single-particle factor accounts for the first remainder; without a single it is absent. By (69), the negative term dominates every fixed multiple of \(\log\mathcal W\). The preceding logarithmic derivative bounds preserve this estimate on a slightly smaller neighborhood. Hence the fraction after \(D_j-1\) in (71), together with the derivatives needed for contraction, is smaller than every inverse power of \(\mathcal W\).

If the right side of (71) is set to zero, then \(Y=0\). After eliminating \(X=DY\), the linearization is triangular: first the \(Y\) equations, then the center equations (66), and finally the equations determining \(D\). Its inverse is bounded by a power of \(\mathcal W\), by Lemma 21. Apply this inverse to the regular system and use its second-derivative row-sum bounds. On a smaller polynomial-radius ball, the resulting Newton correction map is a contraction: its forcing term is superpolynomially small and its Lipschitz constant is less than one. This proves local existence, uniqueness, and the claimed error. The same construction for complex parameters proves local analyticity. Its balls can be chosen to overlap as \(\mathcal W\) varies, because the displacement from the formal solution is much smaller than their radii. Thus the local solutions link along the whole ray.

For fixed \(n_p\), expansion of (66) at large \(\mathcal W\) gives (70); the fraction in (71) is then \(e^{-\mathcal W}\) times a fixed-\(n_p\) polynomial. The only remaining issue is whether an internal singularity could persist identically and prevent use of ordinary Bethe vectors. At formal spacings, the external pair factors cancel in \(A_j\). Its explicit twist factor is \(e^{-2i\chi}\); the remaining factor is independent of the other pair centers and has bounded derivatives, including the single factor if present. At this fixed-\(n_p\) end, the center and single responses to the twists are \(O_{n_p}(\mathcal W^{-1})\), since the eliminated pair equations have the fused phases of (66). Hence an infinitesimal spin twist makes \(D_j-1\) nonzero generically even if it vanishes at zero twist. The exponentially small correction in (71) does not change this conclusion. Analytic continuation gives generic ordinary points along the branch. Finally direct substitution in (57) proves the common translation by \(\zeta/\mathcal W\). At real zero twist, conjugation symmetry, or continuity from the ordinary roots, gives real eigenenergies whenever the vectors are nonzero; no global uniqueness assertion about Bethe roots is needed. ◻

Proof. We first show that the algebraic wave is nonzero and find its limit as \(\mathcal W\to\infty\) at fixed \(n_p\). The positive-overlap criterion will then identify it at that end; simplicity will keep it on the ground level as \(\mathcal W\) decreases to \(W\).

Order the initial momenta in adjacent pairs, lower member before upper member, with the single last if present. We first determine the coefficient produced by the \(B\) operators of Lemma 16. At formal pair spacings, the spin-\(\frac12\) product on a pair, evaluated at its center, maps into the singlet. Its two factors are proportional to an antisymmetric projection and a permutation, which proves this directly. At an arbitrary spectral value the pair product preserves that singlet and acts by the scalar \[(u-u_j)^2+4.\] Consequently, applying \(B(u_j)\) lowers the designated pair into its singlet while acting on previously produced singlets by these scalars. The vacuum-preserving factors on the other pairs can vanish only when pair centers coincide. The real single, if present, remains separated from their dangerous factors by imaginary shifts. Hence the initial coefficient is nonzero and, after division by a scalar, tends to the product of pair singlets in each index. At the spacings (70), this scalar and its reciprocal have fixed-\(n_p\) polynomial bounds. Replacing the formal roots by the true roots changes this normalized coefficient by an exponentially small quantity times such a polynomial.

We must also control coefficients for permutations that reverse a pair. By braid consistency, each can be written as a \(B\) construction with permuted physical sites, multiplied by scalar exchanges on the joint all-up vacuum. Reversing the internal order can cost a factor with denominator \(|X_j-Y_j|\). On the other hand, each single-index \(B\) vector vanishes at the exact string variables when that pair alone is reversed. This can be seen without exchanging its sites. On the all-up vector, the coefficient of the vector with its only down spin at site \(r\) in \(B(a)\) is \[-2i\prod_{s>r}(a-k_s-i)\prod_{s<r}(a-k_s+i).\] Suppose the upper member \(k_p=a+i\) precedes the lower member \(k_q=a-i\), so \(p<q\). If \(r<q\), the first product vanishes at \(s=q\); if \(r\geq q\), the second product vanishes at \(s=p\). Thus \(B(a)\) annihilates the vacuum for any separation of these sites and any values of the other momenta. Since the \(B\) entries commute, apply this \(B(a)\) first in the full spin vector. This proves the claimed vanishing as a polynomial identity, without a potentially singular conjugation.

Apply this vanishing separately to each reversed pair. Taylor’s formula in the corresponding variables bounds each single-index vector by a product of factors \(|X_j|+|Y_j|\). Precisely, for each reversed pair subtract the evaluation at \(X_j=Y_j=0\). These evaluation operators commute because their variable pairs are disjoint, and each subtracted evaluation is zero by the preceding identity, with all other variables unrestricted. The product of the resulting difference operators is the original vector. Applying the integral form of Taylor’s formula to this product gives the claimed product bound, with fixed-\(n_p\) polynomial constants at (70). The two indices supply the square of this product, whereas the vacuum scalar costs only one factor \(|X_j-Y_j|^{-1}\) per reversal. Since \[X_j-Y_j=(D_j-1)Y_j, \qquad Y_j=(D_j-1)\times\hbox{the fraction in \eqref{eq:B17}},\] the quotient is removable even at \(D_j=1\). At fixed \(n_p\) its bound is \[\operatorname{poly}_{n_p}(\mathcal W)e^{-r\mathcal W}\] when \(r\) pairs are reversed. All remaining permutation factors cost only fixed-\(n_p\) powers, because distinct centers are separated. The same argument works with infinitesimal twists and then passes to zero twist. In particular the initial ordered amplitude remains nonzero at the large-circle end, and singular roots there yield ordinary continuation-limit eigenfunctions.

To apply Corollary 20, rescale the circle of length \(\mathcal W\) at coupling \(2\) to a fixed length \(W_0\). The new coupling is \(c=2\mathcal W/W_0\), momenta are multiplied by \(\mathcal W/W_0\), and energies by \((\mathcal W/W_0)^2\). The eigenenergy is \[-Mc^2/2+O_{n_p}(1).\] Consider a coordinate chart with the seam avoided, distinct pair centers separated by a fixed positive distance, and within-pair gaps rescaled by \(c\). Reversed-pair amplitudes are negligible there, because the spans remain short of a full circumference. A permutation that pairs positions near different separated centers is negligible for the same exponential reason. The remaining permutations exchange entire blocks. Their coefficients are given, to leading order, by the two fusion scalars in the line calculation. As \(c\to\infty\), the pair–pair scalar tends to \(-1\), and the pair–single scalar tends to \(1\).

The momentum conjugate to a pair center is twice its individual momentum. Thus (70) gives the center wave \[\det\left[ \exp\!\left\{\frac{2\pi i}{W_0} \left(j-\frac{M+1}{2}\right)q_a\right\} \right]_{j,a=1}^M,\] together with the normal profile (65) and a constant single wave if present. For ordered distinct centers, the Vandermonde formula is a constant phase times \(\prod_{a<t}\sin(\pi(q_t-q_a)/W_0)\). This product is strictly positive. After the static sign change, each double singlet has nonnegative components, as does the possible majority-color single.

The same phase choice applies to charts crossing the seam, by periodicity and translation. To check the latter point, sum (66) to get \(2\sum_j u_j+\ell v=0\). At zero twist, periodicity quantizes the total momentum of a nonzero true wave in \((2\pi/\mathcal W)\mathbb Z\). Its difference from the formal value is \(o(\mathcal W^{-1})\) at the fixed-\(n_p\) large-circle end, so that integer is zero there. The integer is constant along the continuous root branch, including continuation limits, and hence the total momentum is zero throughout.

The convergence just obtained is uniform on compact separated-center charts with bounded rescaled gaps, before the final \(L^2\) normalization. After normalization, the wave on each such chart is a common scalar multiple of this limiting shape, up to a vanishing relative error. The scalar is bounded above because the shape has positive norm on any one nonempty chart. It cannot tend to zero along a subsequence: Lemma 19 would then give zero limit on every retained chart, although those charts retain the full norm. Every normalized limit consequently has the same strictly positive center shape, with the chart cutoffs included. Corollary 20 identifies the plane branch with the ground state for sufficiently large \(c\).

Finally decrease \(\mathcal W\) along the branch of Lemma 23. The energy is locally analytic and is an eigenvalue at generic parameters. The plane coefficients and \(B\) vectors are meromorphic; their nonvanishing at the large-circle end shows that ordinary nonzero states occur on an open dense set. At complementary parameters, take the twist continuation first and then pass to zero twist. Continuity of the compact-resolvent spectrum on a finite circle keeps the limiting energy in the spectrum. The balanced ground energy remains simple by Lemma 11, so a continuous eigenvalue branch already equal to it cannot leave it without a forbidden degeneracy. The identification therefore holds all the way to \(\mathcal W=W\). Commutation and positivity in Lemma 11 now identify the corresponding transfer eigenvalue with \(\|T_{n_p}\|\). The same simple eigenvalue admits the infinitesimal twist continuation used below. ◻

Proof. At an ordinary point of the root branch, choose a momentum ordering whose color coefficient \(A\) in the Bethe wave is nonzero. The transfer preserves its eigenline, so the coefficient of \(\exp(i\sum_a k_aY_a)\) in the transferred wave is \(\Lambda_{n_p}A\). We compute this coefficient, initially at the nonresonant parameters specified below.

We first compute the contribution near one output coordinate \(Y\). Let \(X\) be the integration coordinate assigned to it, let \(k\) be the momentum carried by \(X\), and fix the rotor sector \((q,j)\) on the intervals outside these two insertions. Put \(d=2j+1\) and \(z=ik-q\). If \(S_j\) denotes the spin-\(j\) angular momentum, the operators \(\eta_L=S_j\cdot\sigma\) and \(\eta_R=S_j\cdot\sigma\) act on the left and right rotor and binary indices, respectively. These two operators commute. We extract the coefficient of \(e^{ikY}\) obtained by taking \(Y\) as the integration endpoint for \(X\). The sum of this coefficient over the two orders \(X<Y\) and \(X>Y\) is \[ \frac{-2(z-2\eta_L)(z-2\eta_R)} {\bigl((d+1)^2-z^2\bigr)\bigl((d-1)^2-z^2\bigr)}. \tag{73}\] We verify both the normalization and the endpoint denominators in this formula.

Suppose first that \(X<Y\), with no other insertion between them. Multiplication by the conjugate entry at \(X\) changes the intermediate sector to \((q-1,j\pm\tfrac12)\). Its energy minus \(E_{qj}\) is \(1\pm d-q\). On one matrix index the Clebsch–Gordan projections in \(j\otimes\tfrac12\) are \[P_+(\eta)=\frac{j+1+\eta}{d},\qquad P_-(\eta)=\frac{j-\eta}{d}.\] The two matrix indices supply \(P_\pm(\eta_L)P_\pm(\eta_R)\), with the same choice of sign because they belong to the same intermediate rotor representation. The normalization of the Haar harmonics supplies the additional factor \(d/(d\pm1)\). Indeed, multiplication from a normalized spin-\(j\) matrix element to a normalized spin-\((j\pm\tfrac12)\) matrix element has coefficient \(\sqrt{d/(d\pm1)}\) times the two Clebsch–Gordan coefficients. The return insertion is its adjoint and contributes the same factor. Thus this dimension factor occurs once in the product of the two insertions, not once for each matrix index.

For \(X<Y\), the input plane times the heat factor is \(e^{ikX}e^{-(1\pm d-q)(Y-X)}\). Its upper-endpoint contribution is \(e^{ikY}/(1\pm d+z)\). For \(X>Y\) the lower-endpoint calculation gives denominator \(1\pm d-z\). The exterior transpose and the conjugation by the antisymmetric tensor replace \(\eta_L,\eta_R\) by \(-\eta_L,-\eta_R\) in that contribution. Consequently the full coefficient is \[\sum_{s=\pm1}\frac{d}{d+s} \left\{ \frac{P_s(\eta_L)P_s(\eta_R)}{1+sd+z} +\frac{P_s(-\eta_L)P_s(-\eta_R)}{1+sd-z} \right\}.\] Substituting the two projection formulas and taking a common denominator gives (73). If \(d=1\), the minus representation is absent: we omit it before simplifying. The result is the canceled value of (73), in particular \(2/(4+k^2)\) when \((q,j)=(0,0)\).

We next extract the coefficient of a full output plane \(\exp(i\sum_a k_aY_a)\) in the actual heat-kernel integral. Initially choose a nonsingular Bethe vector and parameters for which all integrated partial-sum rates are nonzero, no \(ik_a\) is a pure rotor-energy difference, and no \(i(k_a-k_t)\), for \(a\ne t\), is such a difference. The relevant energy differences are even integers. These conditions can be met on the branch: a generic common twist \(\zeta\) removes pure-rotor coincidences and zero partial-sum rates. Near the large-circle end, the separated centers in (70) and an infinitesimal spin twist \(\chi\) resolve the remaining pair-difference resonances. Here \(n_p\) and \(c=2\) are fixed while the circumference varies. We will pass to all other parameters by analytic continuation after computing the coefficient.

Fix a total ordering of \(X\) and \(Y\) and expand in rotor harmonics. An ordered block \(x_1<\cdots<x_r\) of input coordinates in a gap \((a,b)\) between consecutive output coordinates has exponential rates \(\alpha_1,\ldots,\alpha_r\). Repeated integration produces terms with endpoint phases \[\exp\left\{a\sum_{h=1}^{s}\alpha_h +b\sum_{h=s+1}^{r}\alpha_h\right\}, \qquad 0\le s\le r.\] Thus a prefix is assigned to the left endpoint and the remaining suffix to the right; this description includes the nested integrations whose intermediate endpoints coincide. Over all gaps it gives a weakly order-preserving assignment of inputs to outputs or to the seam endpoints \(0,W\).

At an output \(Y_a\), the exponent is the sum of the assigned input momenta, multiplied by \(i\), plus the adjacent rotor-energy difference. If no input is assigned there, the exponent is purely a rotor-energy difference and cannot equal \(ik_a\). Every output therefore receives at least one input. Since their numbers agree, exactly one input is assigned to each output and none to the seam. The difference nonresonance then forces that input to carry \(k_a\), and forces its remaining rotor-energy difference to vanish.

In particular, a contributing gap contains at most two inputs, one assigned to each endpoint. When two integration variables lie in the same \(Y\) gap and are assigned to its opposite ends, the coefficient is the product of their two endpoint factors. This can be checked without separating the ordered integrations: for exponential rates \(\alpha,\beta\), the term assigning \(x_1\) to \(a\) and \(x_2\) to \(b\) in \[\int_a^b e^{\beta x_2}\int_a^{x_2}e^{\alpha x_1}\,dx_1\,dx_2\] has coefficient \(-1/(\alpha\beta)\), precisely the lower and upper endpoint coefficients multiplied together. The other terms assign the two variables to a common endpoint and do not contribute to the required plane. After every \(X\) has been paired with a \(Y\), the vanishing energy differences show that all intervening intervals carry the same rotor energy. The charge has already returned to the same \(q\), so equality of energies forces the same \(j\). Thus one fixed through sector \((q,j)\) contributes the ordered product of (73) over the momenta, followed by its seam trace.

To justify coefficient extraction also for the infinite rotor sum, work in the ordered complex tube \(0<\operatorname{Re}Y_1<\cdots<\operatorname{Re}Y_{n_p}<W\). At fixed finite \(n_p\), intermediate labels differ from the through labels by bounded steps, so their energies differ from \(E_{qj}\) by \(O_{n_p}(1+|q|+d_j)\). On each compact subset of this tube, the endpoint terms are therefore bounded by a fixed polynomial in the labels times \[\exp\{-WE_{qj}+C(1+|q|+d_j)\}.\] The quadratic energy gives normal convergence of the analytic exponential sums, including after any finite imaginary translation. Modulo even-integer shifts, their exponent vectors have only finitely many types. The commuting translations \(Y_a\mapsto Y_a+i\pi\) multiply an exponential of rate \(\gamma_a\) by \(e^{i\pi\gamma_a}\); these multipliers distinguish the types. Polynomial interpolation in the translations consequently projects onto each type. In the type containing the required plane, divide by \(\exp(i\sum_a k_aY_a)\). The resulting locally convergent Laurent series in \(w_a=e^{2Y_a}\), on \(1<|w_1|<\cdots<|w_{n_p}|<e^{2W}\), has the desired plane coefficient as its constant coefficient. Uniqueness of Laurent coefficients proves that no other term of the infinite sum changes the extraction above.

Finally, \[i(z-2\eta)=(-iq-k)-2iS_j\cdot\sigma =\mathcal L_j(-iq-k).\] The ordered product in each matrix index is therefore the monodromy whose twisted trace has eigenvalue \(t_j(-iq)\) in (58). The two indices give its square. Multiplying by \(\lambda^{n_p}\) and summing the seam weights yields (72). At zero twist this is the top transfer eigenvalue by Proposition 24 and Lemma 11; exceptional ring lengths follow by continuity. ◻

Proof. Set the spin twist to zero, initially retaining continuation from arbitrarily small nonsingular spin twists where needed. In (58), evaluated at \(-iq\), every shifted argument of \(Q_0\) has an odd integral imaginary part relative to a real formal root. Indeed \(2m\equiv2j\equiv q\pmod2\), so \(q+2m\pm1\) is odd, and the same is true of \(q\pm(2j+1)\). These factors are uniformly separated from zero.

At the formal pair spacing \(k_{r\pm}=u_r\pm i\), the contribution of this pair to \(F\) and its accompanying \(Q_0\) factors in each summand of (58) cancels to \[(-iq-u_r)^2+d^2,\qquad d=2j+1,\] independently of \(m\). It follows that, in every one of the \(d^2\) summands obtained by squaring \(t_j\), the absolute contribution of this pair relative to its \((0,0)\) contribution is \[\frac{(u_r^2+1)(u_r^2+9)} {\left|\bigl((iu_r-q)^2-(d+2)^2\bigr) \bigl((iu_r-q)^2-(d-2)^2\bigr)\right|} \le1.\] To see the inequality, factor the denominator into four linear factors. Their absolute values are \(\sqrt{u_r^2+w^2}\), where the four widths are \(|q\pm(d+2)|\) and \(|q\pm(d-2)|\). All are positive odd integers, and at least two are at least three. Their product is consequently at least \((u_r^2+1)(u_r^2+9)\).

This comparison remains uniform for the true roots. The linear factors just used, and the individual factors before cancellation for the two pair constituents, stay a fixed distance from zero. Equations (69)–(71) make the root corrections smaller than every power of \(W^{-1}\). There are only polynomially many factors in \(W\) in the particle band. Hence the total multiplicative change is bounded, and in fact differs from one by less than every prescribed power of \(W^{-1}\). This is the point at which an estimate per particle would have been insufficient without the sign-free bound of one at the formal spacing.

If \(\ell=1\), denominators contributed by the single momentum can vanish at zero common twist before cancellation. We control them on one complex \(\zeta\) circle instead. Choose a sufficiently large fixed \(C\) and put \(|\zeta|=W^{-C}\). The common twist translates every momentum and auxiliary root by \(\zeta/W\). At \(\zeta=0\) the single momentum is zero up to an error smaller than every power of \(W^{-1}\). Thus on this circle every possibly vanishing single factor has modulus bounded below by a fixed multiple of \(W^{-C-1}\). All single-root contributions are consequently bounded by a fixed power of \(1+W+d+|q|\). The change in all pair factors together is bounded as well: each argument moves by \(W^{-C-1}\) and the number of factors is polynomial in \(W\). Choosing \(C\) once sufficiently large controls their product.

We now justify using the maximum-modulus principle on this circle; this requires analyticity on its whole interior, with no appeal to a lower bound for the gap between different total-momentum sectors. Let \(\psi_0\) be the normalized untwisted ground vector, with total momentum zero, and define the change of gauge \[(U_\zeta\psi)(X) =\exp\!\left(\frac{i\zeta}{W}\sum_{t=1}^{n_p}X_t\right)\psi(X).\] After this change of gauge, the Hamiltonian on periodic functions is \(H_2+2\zeta P/W+n_p\zeta^2/W^2\), where \(P=-i\sum_t\partial_{X_t}\). In the \(P=0\) subspace its ground line is therefore the same simple line for every \(\zeta\). Write \(T_{n_p}(\zeta,0)\) for the twisted transfer. Its rotor expansion is analytic, and so is the fixed-gauge scalar \[\left\langle\psi_0, U_\zeta^{-1}T_{n_p}(\zeta,0)U_\zeta\psi_0\right\rangle.\] Near zero this scalar equals \(\Lambda_{n_p}(\zeta,0)\) by commutation. It therefore continues \(\Lambda_{n_p}\) across any crossings with other momentum sectors. The coefficient formula may first be used with arbitrarily small spin twists and then continued to zero spin twist, equivalently taking the removable analytic limits in (72).

On the \(\zeta\) circle, the ratio of the nontrivial-sector sum to \(Z_{n_p}\) is therefore bounded by \[C\sum_{(q,j)\ne(0,0)} d^2(1+W+d+|q|)^K \exp\{-WE_{qj}+W^{-C}|q|\} =O(e^{-c'W})\] for fixed \(K\) and some \(c'>0\). Every nontrivial admissible sector has \(E_{qj}\ge2\), and the energies grow quadratically in \(q,d\), so the polynomial factors and the number of summands are absorbed by the exponential. The ratio is analytic on the enclosed disk: \(Z_{n_p}\) is analytic and nonzero there, and the whole nontrivial-sector sum is \(\Lambda_{n_p}-Z_{n_p}\). The maximum-modulus principle gives the same bound at zero twist. For \(\ell=0\) the separated pair factors already give the bound directly; the same circle argument also applies.

It remains to take the logarithm of the leading product. Since \(2\lambda=b^2\), a formal pair contributes \[\frac{b^4}{(4+(u_r+i)^2)(4+(u_r-i)^2)} =\frac{b^4}{(u_r^2+1)(u_r^2+9)}=e^{p(u_r)},\] and the possible zero single contributes \(b^2/4=e^{f(0)}\). The cumulative error from replacing the true roots by the centers of (66) is smaller than every power of \(W^{-1}\) by (71); the nontrivial-sector error is exponentially smaller. The positive zero-twist eigenvalue therefore has the logarithm (74). ◻

Proof. For the upper bound, replace each \(\epsilon(u_r)\) by its positive part. Use the counting coordinate \[s(x)=\int_0^x d(y)\,dy, \qquad s(u_r)=\frac{r-(M+1)/2}{W}, \qquad s(\pm B_c)=\pm\frac{M}{2W}.\] The centers are exactly the midpoints of cells of length \(1/W\) in this coordinate. The estimates accompanying (67) give \(d\ge c>0\) and polynomial bounds in \(b\) for its fixed-order derivatives on the needed interval. The integral equation (40) gives such bounds for the fixed-order derivatives of \(\epsilon\). Moreover \(|\epsilon'|\le\pi/2\) on the whole line by Proposition 13. Thus, for the inverse function \(x=x(s)\), \[\left|\frac{d}{ds}\epsilon(x(s))\right| =\left|\frac{\epsilon'(x(s))}{d(x(s))}\right|\le C,\] while each further fixed-order derivative is polynomially bounded in \(b\).

Set \(g(s)=\epsilon_+(x(s))\). On any consecutive collection of full cells in a smooth portion, with outer endpoints \(\alpha,\beta\), the composite midpoint expansion gives \[\sum_{\text{these cells}}g(s_r) -W\int_\alpha^\beta g(s)\,ds =-\frac{g'(\beta)-g'(\alpha)}{24W} +O\!\left(\frac{\operatorname{poly}(b)}{W^2}\right).\] The first derivatives are uniformly bounded, and the higher derivatives and interval length cost only polynomial factors in the remainder. Thus this error is \(O(W^{-1})\) with a constant independent of \(b\). At most two cells meet the cutoffs \(x=\pm B_*\). Since \(\epsilon\) vanishes there and its derivative in \(s\) is bounded, these cells contribute \(O(1/W)\) as well. Consequently \[\sum_r\epsilon(u_r) \le W\int_{-B_c}^{B_c}\epsilon_+(x)d(x)\,dx+\frac{C}{W} \le W\int_{\mathbb R}\epsilon_{\rm in}(x)d(x)\,dx+\frac{C}{W}.\] The last inequality uses positivity: any positive portion outside the counting interval is only omitted on its left-hand side.

For the lower bound choose \(M\) as in the statement. Equation (43) and the relation between \(b\) and \(B_*\) imply \(2\int\rho_{B_*}=b^2+O(b)\), so these particle numbers satisfy the band condition after enlarging its fixed constant. Since \(\int_{-B_c}^{B_c}d=M/W\), (67), including its uniform \(L^1\) bound for the impurity correction, yields \[\left|\int\rho_{B_c}-\int\rho_{B_*}\right|=O(W^{-1}).\] The densities increase with their interval: this follows by positive iteration of their defining equation. Their difference in total mass therefore bounds from below the mass on the excess or missing interval. First \(\rho_B\ge1/\pi\) gives \(|B_c-B_*|=O(W^{-1})\). Those intervals then lie within distance one of the relevant larger endpoint; applying the edge bound \(\rho_B\ge c\sqrt B\) from (43) improves this to \[|B_c-B_*|\le\frac{C}{W\sqrt{B_*}}.\] On this excess or missing interval the counting density obeys \(d\le CB_*\), by (67) and its defining convolution, and \(|\epsilon(x)|\le C\,\bigl||x|-B_*\bigr|\). Therefore replacing the counting interval by \((-B_*,B_*)\) in the dressed-energy integral costs at most \[CWB_*|B_c-B_*|^2\le\frac{C}{W}.\] The same midpoint estimate, now for the smooth function \(\epsilon\) on the counting interval, compares its sum and integral with error \(O(1/W)\). Combining these two bounds proves the lower estimate for the bracket. ◻

Proof. Subtract \(J_s\) from a strip field and expand its zero-endpoint part in sine modes. Completing the square at eigenvalue \((\pi m)^2\) gives (78). At \(z=0\) its quadratic form is \(l\int_0^1|J_s(c,c')|^2ds\), which is positive for a nonzero endpoint pair. The resolvent is positive on \([0,4]\); compactness gives uniform bounds, and the poles stay away from a common neighborhood of that interval. Summing strips gives the stated affine form.

We recall the quantitative free input from (OpenAI 2026b, Section “The linear block transformation” and Appendix “Uniform analytic estimates for the linear block map,” Lemmas “Uniform shells and strips” and “Alias syzygy and coarse division”). If \(b_Q(k)\) is the Fourier symbol of \(Q\), then \[K_{h+1}(p)^{-1}=1+L^{-2}\sum_{l\bmod L} b_Q(k_l)b_Q(k_l)^*K_h(k_l)^{-1},\qquad k_l=(p+2\pi l)/L.\] With stars continued analytically from real momentum, the explicit depth formula has principal numerator \(m^2K_0(p/m)\), where \(m=L^h\). The product of bump symbols has nonprincipal shell sums bounded by \(CA_0^h\). The only estimates on the initial numerator required in that proof are even analyticity on a common strip, bounded derivatives, real ellipticity, and its specified quadratic jet with a fourth-order remainder. Both of our precisions have these properties. In particular, on every nonprincipal alias \(p_l=p+2\pi l\), \[|m^2K_0(p_l/m)|\ge c(1+|l|)^2, \qquad |m^2(K_{0,1}-K_{0,2})(p_l/m)| \le C|p_l|^4/m^2.\] Real ellipticity proves the first estimate on the real cube; reducing the imaginary width preserves it. Even Taylor expansion on a larger analytic cube proves the second. The reciprocal difference is therefore \(O(m^{-2})\). Summation with the product-bump weights costs \(CA_0^h\), and \(L>A_0\) gives \(CL^{-h}\). The principal alias has its direct Taylor estimate. The principal denominator stays bounded away from zero on a common smaller strip, so Cauchy estimates give every fixed derivative, including the shape derivatives.

For clarity, the other free conclusions we use are \[\begin{align*} &C_h=(K_h+Q^*Q)^{-1},\qquad K_h\mathsf P_h=Q^*K_{h+1},\qquad I-Q\mathsf P_h=K_{h+1}, \\ &\mathsf P_h1=1,\qquad \mathsf P_h(\text{centered affine field}) =\text{its affine extension},\\ &\sup_x\sum_y e^{c|x/L-y|} |\nabla_f^s\mathsf P_h(x,y)|\le C_sL^{-s},\qquad s=0,1,2. \tag{79}\end{align*}\] The covariance has exponentially decaying kernels in block units, with constants allowed to depend on fixed \(L\).

Here is the edge form of the positive completion imported from (OpenAI 2026b, Lemma “Positive completion of the block transformation”). Choose a common \(c_0>0\) and write \(\mathsf A_h=(\sqrt{c_0}I,\ell_h)^\mathsf t\), so that \(K_h=d^*\mathsf A_h^*\mathsf A_hd\). There are operators \(\mathcal M_h,\mathcal N_h\) with \[\begin{align*} \mathcal M_h^*\mathcal M_h+\mathcal N_h^*\mathcal N_h&=I,\\ \mathcal M_h^*(\mathsf A_hdf,V-Qf)&=\mathsf A_{h+1}d'V,\\ \mathcal M_h\mathsf A_{h+1}d' &=(\mathsf A_hd\mathsf P_h,I-Q\mathsf P_h),\qquad \mathcal N_h\mathsf A_{h+1}d'=0. \tag{80}\end{align*}\] The part of \(\mathcal M_h\) taking coarse data to fine edge rows has weighted row bound \(C/L\) and column bound \(CL\); its first two fine differences have bounds \(C/L^2,C/L^3\). The coarse parts have bounded row and column sums. All these statements retain the history gain.

For our shapes, start the raw edge lift at the shape diagonal rather than the identity. Division of its fourth-order-zero remainder and the curl addition used in the cited proof remain uniformly bounded. The two principal linear jets in the construction of the edge intertwiner have the same shape factor; hence both alias divisions and their gluing apply. The common real bounds give positivity. At depth zero retain the specified strip lift without adding the fine inverse-curl correction; deleting that correction improves the positive-defect inequality in the cited proof. Its second-stack rows are the square root after removing \(c_0/2\) on each side of (78). Zero rows can be added to give common index sets. Finally take the direct sum of the \(q\) and \(w\) constructions, multiplying all \(w\) edge, observation, and propagated output entries by \(\sqrt\chi\). This proves the assertions for the four kinetic coefficients. ◻

Proof. For the quotient metric \(d\), put \[A_Y=\sum_{x,z\in Y}Q_Y(x)Q_Y(z)d(s_x,s_z), \qquad s_x=(q_x,w_x).\] The medoid \(m_Y\) satisfies \(\sum_xQ_Y(x)d(m_Y,s_x)\le A_Y\): average the same expression over candidate constituents. The triangle inequality therefore gives \[d(m_Y,m_{Y'})\le A_Y+A_{Y'}+ \sum_{x\in Y,z\in Y'}Q_Y(x)Q_{Y'}(z)d(s_x,s_z).\] Bound each distance by a coordinate-path sum of \(r_e\). Smooth bump weights give total path congestion at most \(C/L\) on an edge, proving the first term in (85). The argument works before quotienting on a flat patch. Normalizing a quaternion mean costs only a fixed factor when \(|m_Y|\ge1/2\); on its fallback branch the weighted path bound is already bounded below by a positive constant. Add the two noise distances to finish the estimate.

We describe the conditional change of variables, since it explains both the phase and the absence of an extra period factor. Fix a tree in each fine block. Site shifts \(n_x\) set all its tree-edge integers to zero; the remaining integers lie off the tree. Align the observation in this lift, using the lifted medoid on a curved block. Tree equations fix the differences of the \(n_x\), leaving one root shift. Summation over observation sheets supplies that shift. Equivalently, choose the root representative so that the observed real coordinate lies in the basic interval; conditioning this coordinate to be \(Tw'_Y\) unfolds the root, and then all tree coordinates, over \(\mathbb R\). Reducing the fine coordinates reconstructs the original integers and observation sheet, so this is a bijection. Its only output scaling Jacobian is \(T\) per block.

On a flat adjacent pair the residual integers inside both blocks vanish, and all fitted seam integers equal the prescribed coarse integer. On a curved pair the least-distance condition reads only the quotient output and its specified integer. The tree used inside a curved block cannot enter a flat-pair rule at that block. Thus tree choices do not alter the identity; they may be symmetrized identically in both inputs.

Summing temporal real differences in these tree coordinates telescopes. Since each coarse temporal edge has \(L\) crossing fine edges and \(P_j=TP_{j-1}\), \[\sum_{e_t}u_e-LT\sum_{E_t}u'_E =P_j\left\{\sum_{\text{internal }e_t}k_e+ \sum_{E_t}\sum_{e\text{ crossing }E}(k_e-k'_E)\right\}.\] Internal residual integers vanish on a flat block, and every crossing difference vanishes on a flat pair. Multiplication by \(i\theta\) gives the claimed local phase factors. They are independent of continuous field variations in fixed sheets. ◻

Proof. The direct \(\mathcal F\) row pays \(ct^2\) at an edge seed and \(ctr_e\) at a true input flag. Its cross term has coefficient \(Ct/L\). A failed second-stack mask can be charged to one witness \(r_e>t\). A witness hits at most \(C(1+\log(r_e/t))^2\le C'r_e/t\) rows; the extra \(D_0\) exclusion only changes this constant. Exponential column sums therefore bound the total loss by \[\frac{Ct}{L}\sum_{r_e>t}r_e.\] Large observation displacements have their square reserve, after the fixed noise threshold is increased. The direct \(\mathcal C\) row pays \(ct'r'_E\) on an output flag and is zero otherwise; its second-stack part cancels.

On a flat stencil, the free identities give the quaternion defect bound \[C_Lt e^{-c_LM_j} +Ct\sum_Y e^{-c\operatorname{dist}(E,Y)}|m_Y-\widehat m_Y|.\] The real-coordinate identity is linear and exact before truncation. After truncation, subtract a common real constant on the enlarged stencil. Differences of coordinate values cost at most its length times the largest increment or noise there. The error is therefore \[C_LM_j^Dt e^{-c_LM_j} \bigl(1+\max_{\text{enlarged stencil}}(r_e,|\eta_Y|)\bigr).\] This estimates coefficients of finite stencils, not remote unbounded values. A maximum above the default threshold lies in the same core and is paid by its direct or noise reserve. For the quaternion mean, the block path estimate gives \((C/L)\sum r_e\); on blocks with chords below \(T_j\) its error improves to \(C_LT_j^2\). Every other block has a witness above \(T_j\), to which the \(Ct/L\) charge is assigned.

It remains to estimate a stencil that sees curvature, where a free differential identity is unavailable. Its \(\mathcal Z\) row costs \(C_Lt^2\). For \(\mathcal U\), use the absolute coefficient envelopes in (90). The fine part of \(\mathcal M_s\) has row sum \(C/L\), and the input stack has bounded column sums. Summing curved rows consequently gives a fine-input charge \[\frac{Ct}{L}\sum_e a_er_e,\] where the nonnegative incidence weights have bounded column multiplicity. The observation and coarse-stack column sums similarly give \(Ct\sum_E a'_Er'_E+Ct\sum_Ya_Y|\eta_Y|\). Substituting (85), each neighboring block pair uses an input edge only a bounded number of times, so the coarse charge again gives \((Ct/L)\sum r_e\) plus noise terms. The constants multiplying \(L^{-1}\) are independent of \(L\). Large increments are paid by the direct rows; large noises are paid by their squares. All remaining increments, noises, and \(\mathcal Z\) rows cost at most \(C_LM_j^Dt^2\) per curvature witness. Such a witness has \(r_e\ge P_j/4\). The ratio of this last cost to \(tP_j\) is \(C_LM_j^Dt/P_j\), which tends to zero. Output curvature also forces fine curvature in the enlarged stencil: the pair-lift rules carry a flat neighborhood to a flat neighborhood, including across a four-block corner.

Choose \(L\) to spend less than a fixed fraction of the direct reserve on all \(L^{-1}\) charges. Only flat small-angle stencils are used for selected affine squares; their subtraction costs \(C_LM_j^Dt^3\) per row. Frozen angular extensions are cut off at \(C_Lt\). Exterior principal logarithms retain the chart and alignment protectors, with alignment imposed also on the real coordinate. The remaining losses are smaller than the reserve once \[tM_j^D\ll1, \qquad M_j^D(T_j^2/t+e^{-c_LM_j}/t)\ll1.\] These choices leave (91) and a fixed \(ctP_j\) whenever there is curvature. On a no-output-flag extension through \(r'_E<4t'\), the least-distance output integer, if queried, is unique; a flat-pair constraint is a fixed-sheet constraint. The same estimates thus hold on the required derivative supports. ◻

Proof. The abstract Gaussian estimate is (OpenAI 2026b, Appendix “Joint majorants for Gaussian prepared factors,” Lemma “A joint Gaussian product estimate” and Corollary “Fixed-order derivatives and an exceptional reserve”). We recall its short argument. If \(|Q_\alpha|\le e_\alpha U_\alpha\) and \(U_\alpha\ge1\), then \[|e^{Q_\alpha}-1|\le\sqrt{e_\alpha} e^{2\sqrt{e_\alpha}U_\alpha}.\] When \(0\le C_s\le c_*I\) and \(\sup_x\sum_{\alpha:x\in I_\alpha}\sqrt{e_\alpha} <(64c_*)^{-1}\), the Gaussian determinant identity controls the fourth moment of the product of these envelopes. The individual smallness condition \[(2+4c_*|I_\alpha|)\sqrt{e_\alpha} \le\tfrac14\log(e_\alpha^{-1})\] gives \(\|\prod_\alpha(e^{Q_\alpha}-1)\|_4 \le\prod_\alpha e_\alpha^{1/4}\). Hölder with exponents \(4,4,2\) combines the score, ratio product, and simultaneous-failure indicator. A fixed number of derivatives has polynomially many placements in total load. Pay its negative power of \(g\) from one retained reserve; weakening the other reserves uniformly gives the asserted bounds. For a discrepancy, telescope into one marked factor and use its replacement envelope after factoring out the discrepancy size. No bound relative to the old factor is used; that factor may vanish.

The eighth-root sums follow with their support prices, not merely from an unweighted chain sum. The number of literal paths with \(r\) numerical hops is at most \[\operatorname{poly}(M_j,r)(C_LM_j^D)^{r+O(1)}.\] Hölder on this finite path set and the chain bound with eight times the requested exponent give \[\sum_{\alpha:\,r\text{ hops}}e^{As_\alpha}e_\alpha^{1/8} \le\operatorname{poly}(M_j,r) e^{-c_LM_jr/8+O_L(r\log M_j)+CAr},\] which is summable and small for large \(H\).

Exact factorization for disjoint complete supports is (OpenAI 2026b, Appendix “Local sector factors and their complete supports,” Lemma “Local marginals and factorization”). Its hypotheses are precisely the inclusion of all window and hole queries and all positive and contrary status reads. They make the frozen sets disjoint, the retained marginal covariance block diagonal, and the window data unchanged when the other factor is omitted. This is an assertion about these marginal laws, not conditional independence under an unspecified Gaussian law. ◻

Proof. The principal terms involve at most four origins. Strengthen their masks to one common ball and use an even fluctuation guard. The shared Gaussian tail pays the complement. Count each relative output field and each \(gZ\) as one power of \(g\), and each kinetic prefactor \(B\) as \(g^{-2}\). The vertices are the nonlinear mean, actual row minus affine square, stationary square, old polynomial, and Haar Jacobian vertices of (OpenAI 2026b, Section “The canonical coefficients”), with the real coordinate in direct sum and the additional input cubic vertex. Taylor’s theorem through normalized order four gives remainder \(C_Lg^5\operatorname{poly}(M_j,p_j)\), including normalized derivatives and paired estimates. Restoring full Gaussian polynomial moments and full kernels costs exponential accuracy in \(M_j\). Connected Wick coefficients converge in the same exponential path norm: each covariance connection and prediction carries the exponential moments of Lemma 29, and all paths are retained.

A connected term with \(v\) prefactored vertices has at least \(v-1\) intervertex contractions. Each contraction costs \(g^2\); hence at most one \(g^{-2}\) prefactor remains. At normalized order \(r\le4\), the maximal field degree is \(r+2\). The possibilities are therefore \(BP_3\) at order one, \(BP_4,I_2\) at order two, \(BP_5,I_3\) at order three, and \(BP_6,I_4,B^{-1}J_2\) at order four, besides scalar and linear terms. Linear \(w\) terms occur at orders one and three. Spatial reflection leaves only the temporal direction, and conjugation makes their coefficients imaginary. The affine quadratic symmetry gives exactly the four Hessians already displayed.

Remove those Hessians at orders two and four. Their coefficients are \(\alpha\) and \(\kappa/B\) in (86). A removal changes the shape at which the next free kernel is evaluated. Accordingly subtract the Taylor expansion of that changed kinetic term through order four: the quartic chart-degree response times \(\alpha\), and, at quadratic degree and order \(B^{-1}\), the second parameter derivative applied to the square of the kick, are both included. The aggregate affine Hessian is linear in the four coefficients, so these corrections remove exactly the intended Hessian. Rewrite slot prefactors at the new \(B\), including the lower-order prefactor change of \(BP_3\), and convert real-coordinate powers by \(T\). This prescription defines the finite Taylor map without an implicit choice of coupling convention. Finally compensate each remaining quadratic orbit by its directional nearest-edge tags.

For the diagonal response within the same slot and at its displayed prefactor, an old coefficient transfers only by linear prediction. A relative coordinate over displacement \(r\) becomes \[\sum_z[\mathsf P_h(o+r,z)-\mathsf P_h(o,z)]\,y_O(z), \qquad O=[o],\] and the same formula holds for the real variable. The first difference bound gives \(C|r|/L\) with the spare exponential path weight. A degree-\(n\) label therefore gains \(L^{-n}\); summing the \(L^2\) fine anchors gives \(CL^{2-n}\). For a quadratic packet, expand each difference into its linear affine part and a second-difference remainder. The product of the two affine parts cancels before absolute values are taken, by the packet’s affine normalization. The remainder costs \(L^{-3}\), hence \(C/L\) after anchor summation. The cost of output compensation is bounded by the path’s second moment. This is the proof of (OpenAI 2026b, Appendix “A fixed coefficient norm for the diagonal canonical map,” Lemma “One norm and constants chosen before the block size”), with directional tags in place of a single isotropic tag. Its hypotheses are precisely (79) and affine cancellation, which hold here.

The factors \(T^n\) are near one. Contractions lower degree and can feed only later slots at a given order; nonlinear products have larger normalized order. Shape and prefactor subtractions respect this triangular order, because prediction preserves affine fields and hence the diagonal quadratic response has zero bulk affine Hessian. All nondiagonal bounds are finite for fixed \(L\). Choose caps successively, including the finite initial caps from Section 8, and then triangular discrepancy weights. Taking \(L\) large gives the strict diagonal contraction; free-history and parameter differences have the stated factor by Lemma 29. ◻

Proof. Choose \(\nu<\nu'<1/2\) with \(4m_0(\nu'-\nu)>1\). First-level increments have Gaussian tails at scale \(2^{-n/2}\). Second-level increments are second Gaussian chaoses, including their deterministic symmetric part; their square roots have Gaussian tails at the same scale. At the \(\nu'\)-Hölder normalization the tail at level \(n\) is bounded by a constant times \[\exp\{-cD^2 2^{(1-2\nu')n}\}.\] Summing over the \(2^n\) intervals first and then over levels proves a Gaussian tail for the corresponding dyadic supremum. The condition on \(m_0\) sums the extra weights in (98). Dyadic concatenation then controls arbitrary increments and gives the asserted rough-path bound. Subtracting the linear endpoint interpolation gives the bridge estimate.

For a deterministic path \(h\), the first-level increment changes by at most \(\|h'\|_\infty|I|\). The new second-level terms are bounded by this quantity times the oscillation of \(X\) on \(I\), together with a term \(C\|h'\|_\infty^2|I|^2\). These estimates are summable with the same weights. Under \(dX\mapsto O(s)dX+h'(s)ds\), subtract on each interval the frozen value of \(O\) times its first increment and the corresponding tensor square times its second increment. Integration by parts with the smooth coefficient gives an additional factor \(|I|\) in the remainders. This also proves convergence of the differentiated dyadic tests. The same estimates apply at every fixed derivative order.

The extension theorem for geometric rough paths (Lyons 1998, Theorem 2.2.1) bounds higher iterated integrals by the usual factorial estimates. Applied to the linear quaternion equation, it gives a convergent series in the path scale. The series is also analytic in added constant-velocity parameters, by augmenting the path with time. On a set where \(g\mathcal N\) is sufficiently small, the quaternion and circle displacements are bounded by \(Cg\mathcal N\). Outside that set compactness gives the needed bound for the quaternion displacement; the circle displacement remains linear. These statements can equivalently be obtained by dyadic polygonal Brownian approximations. Their Stratonovich lifts converge almost surely in Hölder norm and have a common rough-norm bound with a Gaussian exponential moment (Friz and Victoir 2005, Propositions 19 and 26, Corollary 27). Continuity of extension (Lyons 1998, Theorem 2.2.2), together with the factorial bounds, then gives convergence of every fixed iterated integral in every finite \(L^q\).

The heat-kernel Gaussian bound on the fixed compact sphere, multiplied by the real Gaussian density in the circle lift, gives (99); chord and geodesic distances are comparable in the required direction. For (100), split the row at half-time. A large whole-path rough size forces a large size on one half, up to a fixed factor, by concatenation. For the first half, use its Gaussian tail and the heat-kernel supremum on the other half. For the second half use heat-bridge reversal. Stratonovich anti-development reverses with the path, so the same estimate applies.

We spell out the endpoint differentiation, since the unnormalized measure is important here. To vary its endpoints, multiply the quaternion path on the left by a smooth deterministic path \(H(s)\) whose endpoint values make the prescribed changes. Translate the circle path in the same way. The driving increments become exactly \[dX^q\longmapsto \operatorname{Ad}_{H(s)}dX^q +\frac{H'(s)H(s)^{-1}}{\sqrt r\,g}\,ds.\] The rotation is deterministic and orthogonal. Its composition with the deterministic drift changes Brownian measure by the Cameron–Martin exponential (Cameron and Martin 1944, Theorem 2). Differentiating that exponential gives scores bounded by fixed polynomials in \(\mathcal N\) and \(g^{-1}\). The preceding transformation estimates control the differentiated path factors and profiles. Haar and Lebesgue endpoint translations have Jacobian one. Hölder’s inequality, using a sufficiently small part of the Gaussian reserve for the scores, leaves the asserted tail bounds. Off the path-size profiles, the deleted-row factors use the paths only through \(H(s)q(s)\) and deterministic circle translation, whose derivatives are bounded; no small-path expansion is used there.

These identities can first be read as identities of conditional densities for almost every endpoint. For completeness, choose endpoint transformation parameters in a small chart. Fubini gives a base endpoint for which the identities hold for almost every parameter. The transformed integral is smooth by dominated differentiation, using (100) and the Cameron–Martin exponential. It consequently supplies a smooth version throughout that chart. The versions agree on overlaps. This proves the derivative statement for the unnormalized conditional measures, including on closed profile derivative supports. ◻

Proof. The first-order tangent of the developed endpoint is \(A\), because \(Z(1)=0\). The iterated-integral bounds show that its derivative with respect to \(A\) is uniformly close to the identity when \(gp\) is small. The inverse-function estimate applies on a ball containing the entire profile. In that profile \(|a|\le Cp\) and \(\mathcal N(Z)\le CD_sp\), so every possible solution lies in this ball and the inverse is unique there. One may use a fixed larger ball of the same form for the prepared smooth extensions.

Changing the Gaussian endpoint variable \(a\) to \(D\) gives the determinant \(\det(\partial_D A)\) and the exponential \(e^{-|A|^2/(2rg^2)}\). Converting logarithmic coordinates to Haar endpoint density divides by \(J(D_q)\). Separating its value at zero and the leading Gaussian gives (101). The scale and coordinate-normalization factors are exactly \(C_g\). The determinant is positive on the chosen inverse chart. All derivatives follow from the analytic inverse and Lemma 37.

Reversal can be imposed on this rule: reverse \(X\) with its origin subtracted, reverse the bridge, and use the averaged vertical convention already specified. The dyadic size test is invariant under reversal. Thus neither the disintegration nor its extension breaks a required symmetry. ◻

Proof. We first choose a Gaussian law with finite-range covariance and retain its exact density ratio. We then identify its stationary energy and scalar determinant, and verify that profile failures still have a Gaussian reserve under this law.

The numerical Gaussian law. In normalized sine coefficients write \(Z_m=\xi_m/(\pi m)\); the \(\xi_m\) are independent white normals, with four colors at every regular row. The precision and linear term of the completed energy are \[A_m=I+\frac{r^2}{(\pi m)^2}d^*d, \qquad |(L_m)_y|\le \frac{Cp}{m^2}.\] The bound on \(L_m\) follows because the sine coefficient of the affine interpolation is \(O(m^{-1})\). Shrink the fixed neighborhood of \(r=1\) so that \(4r^2/\pi^2<\delta<1\) there. Approximate \(A_m^{-1}\) by the Neumann polynomial \(C_{m,S}\) cut off at a sufficiently small fixed multiple \(S\) of \(M\). On each real spectral value \(x\in[0,\delta/m^2]\), \[C_{m,S}=\sum_{j=0}^{S}(-x)^j, \qquad c\le C_{m,S}\le1, \qquad |A_mC_{m,S}-I|\le (\delta/m^2)^{S+1}.\] These scalar estimates give the operator bounds also on arrays with cuts. In absolute row sums the same Neumann series is dominated by \(\delta/m^2\). It gives exponential localization in the vertical position and a summable \(m^{-2}\) deviation from identity per row.

Set \(\mu_m=-C_{m,S}L_m\) and \(\eta_m=\xi_m-\mu_m\). Use the Gaussian probability with covariance \(C_{m,S}\) and mean zero for \(\eta_m\). This change is exact when accompanied by the factor \(\det(C_{m,S})^{1/2}\) per color and the exponential with exponent \[\frac12\eta_m^*(C_{m,S}^{-1}-A_m)\eta_m -(A_m\mu_m+L_m)^*\eta_m.\] The shift of the mean contributes \[\frac12\mu_m^*A_m\mu_m+L_m^*\mu_m.\] Adding this contribution to the unchanged endpoint quadratic terms in (102) and the horizontal energy gives the numerical stationary energy. The determinant, stationary energy and ratio exponent together make the change of Gaussian law exact.

These changes may first be performed with finitely many bridge modes. The covariance perturbations have summable trace per site and the means are square summable, so the Gaussian laws and their density formulas have limits as the number of modes tends to infinity. The limiting path laws are locally absolutely continuous with respect to independent bridges.

Local factors from the exact ratio. We detail the locality of these corrections. A residual \(A_mC_{m,S}-I\) has bandwidth \(S+1\) and size \((\delta/m^2)^{S+1}\). Expand inverse and determinant ratios in this residual, inserting a local \(A_m\) where needed. A chain of \(h\) hops can be grouped with its whole interval window and assigned the bound \[m^{-2}(Ce^{-c_LM})^h,\] with fixed \(M,p\) powers for its arguments. There are only \(\operatorname{poly}(M)\) choices at each hop, and its full window has load \(C(h+1)\). Group all modes of a given window in one factor. The resulting exponential envelopes involve \[1+\sum_{m\ge1}m^{-2}|\eta_m(y)|^2\] at their read endpoints. More explicitly, for a grouped window chain \(\gamma\) of \(h\) hops we may take a single envelope coefficient \(e_\gamma\le\operatorname{poly}(M,p)(Ce^{-c_LM})^h\); the mode weight \(m^{-2}\) stays inside the displayed quadratic envelope. Its eighth-root support-hit sum is bounded by a sum of the form \[\sum_{h\ge1}\operatorname{poly}(M,p,h) (C M^D)^h e^{AC(h+1)}e^{-c_LMh/8},\] which is small for large \(H_N\). In particular, no eighth root is taken separately on the summable mode weights. The determinant estimate for products of exponential-minus-one factors now applies first in finitely many modes. Its small per-site endpoint load and trace hypotheses hold uniformly because \(\sum_m m^{-2}<\infty\). Passage to all modes gives the same joint bound, including the required eighth-root summability. There is no factor proportional to the number of bridge modes.

The scalar and stationary energy. For the determinant of \(A_m\) itself, group its convergent log series by length and diagonal anchor. Terms below the numerical cutoff are extracted locally; beyond it they have the exponential window prices just established. Use the value on the array without holes for the bulk scalar, and retain the local changes near holes in the compulsory factors. The exact full completion of (102) gives (78). Numerical stationary coefficients and the prescribed polynomially truncated square-root rows differ from it by exponentially small weighted row sums, also on cut arrays. Assign each coefficient difference its endpoint and stencil reads. This gives the claimed local error in chart norms.

Profile failures under the numerical law. The numerical mean paths have \(\mathcal N\) size \(Cp\) and bounded first derivative of that size, since the mean sine terms and their first derivatives are absolutely summable. For a specified set of rows, the centered numerical Gaussian marginal has density bounded above by \(\exp(C\,\#\text{rows})\) relative to the independent bridge marginal: its covariance is at most identity, and the lower and trace bounds control its determinant. Consequently, a failure of a regular-row profile forces the centered path to have size at least \(cD_sp\), also on derivative supports, once \(D_s\) is large. Here the mean bound, chart extension through \(4t_*\), and the inverse in Lemma 38 are all used. Lemma 37 therefore gives the shared Gaussian-tail estimate required in the joint bound. Extend ordinary exponent vertices with larger smooth path-size profiles before opening their failure factors. Outside that extension the good profile is zero, so the failure has the same reserve. Profiles attached to compulsory factors are retained on all their regular-row reads. ◻

Proof. Let \(n_c\) be the number of primary events in the component. Choose its fresh record with \[s_c\le C n_c.\] Indeed, each primary carries only fixed-radius halos in units of \(M\), and a spanning tree of the proximity links joins their footprints by paths of length \(O(M)\) per link. Every halo and joining path therefore has bounded load. We use this particular record, without arbitrarily adding unused load. Consequently a reserve \(cp^2\) per primary gives \(cp^2s_c\) after decreasing \(c\).

First consider only the deleted integrations and their touching interactions, with the normalization \(C_g\) removed. A horizontal flag has the reserve \(cb r_e^2\) from (99). A stochastic deletion has the reserve \(cD_s^2p^2\) from (100). For a vertical flag both its row paths are deleted. The real interaction loss is \[1-\cos U(s)\,q(s)\cdot q'(s),\] one half of the squared normalized matrix-spin chord. At the flagged endpoint the principal vertical branch makes this at least \(cr_e^2\). The triangle inequality for the matrix-spin chord gives the lower bound \[1-\cos U(s)\,q(s)\cdot q'(s) \ge cr_e^2-Cg^2(\mathcal N_1^2+\mathcal N_2^2),\] using the path displacement estimate and decreasing \(c\). Use only a sufficiently small fixed fraction of the interaction loss for this payment. Its adverse term then has an exponential moment controlled by Lemma 37. A fixed Hölder inequality retains positive fractions of the horizontal and stochastic reserves. Each row touches only boundedly many bonds, and if both endpoints are flagged their payments share the same loss. All other touching interactions have modulus at most one. Endpoint derivatives are treated by the last part of Lemma 37; their fixed powers of \(g^{-1},M,p\) are charged to the displayed entropy term.

The direct extracted energy (81) uses only a small fraction of these reserves. At flags it is linear in the large increment, while the payment is quadratic. At nonflag bonds incident on a deletion, the direct squares absent from the regular array cost at most \(Cc_0bt_*^2\) per bond. Their number is a fixed multiple of the number of primary events, and the vertical direct energy has already been split between its two strips. Choosing \(c_0,d_0\) small leaves a positive reserve. Chord/logarithm differences on included direct rows have the Taylor bounds already established.

It remains to pay for the masked second-stack rows. A row whose mask is off extracts no energy. For a row whose mask is on, both the base row and the regular-array row have size \(Ct_*\). If its center has distance \(a\) from the nearest primary, their difference, in a common local frame, is bounded by \[ C\operatorname{poly}(M)t_*^2+Ct_*e^{-ca}+Ct_*e^{-c_LM}. \tag{103}\] To prove this, use a flat chart through a fixed fraction of that distance, within the stencil. The chord/logarithm difference gives the first term. The kernels on arrays with and without holes agree up to their exponential tails at that distance, giving the second; numerical truncation gives the third. Farther increments of a base row can be large, but its mask permits growth only with the prescribed exponential profile, slower than the row decay when \(R_*\) is chosen large. Regular-array increments are all chart sized. For small \(a\) the separate row bounds give the same estimate. Beyond the assigned core ranges this distance comparison is unnecessary because the ordinary local factors are used instead.

Using \(\bigl||a|^2-|b|^2\bigr|\le(|a|+|b|)|a-b|\) and summing the distance term in (103) costs at most \(Cbt_*^2e^{-c'D_0}\) per flag: its masked-on rows begin at distance \(D_0\). It costs at most \(Cbt_*^2\) per stochastic primary. Choose \(D_0\) large and then \(D_s\) large to pay these costs with their respective reserves. On a no-flag extension only stochastic primaries occur, so the same argument applies. Finally, the Taylor and numerical errors near a component cost at most \[CbM^D(t_*^3+t_*^2e^{-c_LM})\] per primary. Since \(t_*\asymp gp\), these leave a positive fraction of \(p^2\) after the stated parameter choices. The support and integration costs are bounded by the displayed \(C_LM^D(1+|\log g|)s_c\). ◻

Proof. On flat cochains the observation preserves the lifts and decouples the quaternion and real-field integrations. To see that the displayed affine coefficients are independent of the particular retained-density representation, fix an untwisted output near the identity with integers zero. Perturb a single quaternion Cartan direction, or the real field, by sine and cosine waves in one lattice direction. On sufficiently small perturbations every mask is one and the covering sum is one. The phase sum is constant. Divide the second variation of the negative log density by volume and by the mean square bare wave gradient, averaging the sine and cosine versions.

As the wave momentum \(p_{\mathrm{mom}}\) tends to zero, this quotient converges to the corresponding displayed affine coefficient. Indeed, at an anchor subtract the common field value. The remaining wave is its affine part plus a second-difference remainder. In the weighted direction norm, these have sizes respectively \(C_L|p_{\mathrm{mom}}|/t\) and \(C_L|p_{\mathrm{mom}}|^2/t\). The canonical compensations and regular errors have zero affine quadratic jet, so their contribution to the quotient tends to zero. Their anchored derivative sums converge by their defining norms. Winding errors at the lowest nonzero momenta have exponentially small period weights, and their direction norms require only the wave bound itself. The kinetic rows have their exact affine normalization and uniform row moments. In the quaternion calculation there is no acceleration term from subtracting the anchor rotation, because the wave remains in one Cartan direction. Increasing dyadic square tori suffice, and the convergence is uniform in the retained class at each fixed layer.

For the flat reference at locally trivial output winding, fixing the output variables and integers fixes the common integration sheet. The reverse-tree parametrization of the observation leaves a genuine real Gaussian integration, with no additional periodization of its zero mode. The local output log density is therefore the quaternion endpoint log of (OpenAI 2026b) plus the log of the observed real Gaussian field. The quaternion observation, including its fallback branch, is exactly that of the cited evolution; induction identifies its coefficients. Gaussian observation preserves affine stiffness in the physical coordinate. The intrinsic characterization just proved shows that different cutoff representations do not affect either conclusion.

For continuity, first work on a fixed finite torus and at fixed output arguments. Brownian initial densities vary continuously with the heat time: use heat-bridge continuity, or finite time-slicing and the heat bounds of Lemma 37. The potentials are continuous and deck sums have Gaussian domination. Sheet predicates are held fixed. The block kernels preserve continuity by dominated convergence; their geometric and medoid predicates do not depend on the varying coupling parameters. The direct-energy and covering bounds provide domination. On the all-good domain the retained representation is a nonzero exponential. Its finite-period log ratios have uniformly bounded higher derivatives, since the possibly large global phase is constant there. Finite-difference approximations to their second variations have a uniform error by these higher-derivative bounds; hence those second variations are continuous. Taking the uniform small-momentum limit identifies the next affine coefficient continuously. Induction proves the statement at every finite admitted depth. No bound on a derivative of the total accumulated phase is needed. ◻

Proof. Parameterize the initial values by the four uncorrected discrepancies \(x=(x^q,x^w)\): \[x^q=b(r^{-1},r)-(\beta,\beta),\qquad x^w=R_N^{-2}\{b(r^{-1},r)-(c_x,c_t)\}.\] Every fixed ball in these coordinates is available for sufficiently large \(\beta\). Indeed the first two coordinates determine \(b\) and \(r\) by product and ratio, and the last two then determine \(c_x,c_t\). Proposition 36 gives \(|\lambda_N-x|\le C_L\), while all initial distance terms are uniformly bounded.

Choose \(\rho\) as stated and let \[z_h=\rho^{-h}\lambda_{N-h},\qquad 0\le h\le N.\] Run each shot until its first possible exit from \(|z_h|\le D\). Convolution of the first estimate in (87), followed by the second, gives on every such segment, including its first attempted output, \[\begin{align*} U_{N-h}&\le (C'_L+C'_LH^{-c}D)\rho^h,\\ |z_{h+1}-\rho^{-1}z_h|&\le C''_L+C''_LH^{-c}D. \end{align*}\] Choose \(D\) large and then \(H\) large so that the right side of the second inequality is much smaller than \((1-\rho)D\). The comparison with the reference then puts the entire barrier, and the next coefficient step, strictly inside the admitted bands.

We give the topological argument explicitly. Use the initial \(x\)-ball of radius \(D+C_L+1\). Suppose no admitted shot in this ball has zero terminal discrepancy. If its initial discrepancy is outside the barrier, map the shot to its radial direction. Otherwise join successive \(z_h\) by line segments and use the direction at its first crossing of \(|z|=D\). If it has no crossing, use the direction of its nonzero terminal value. This defines a map from the whole parameter ball to the unit sphere.

The map is continuous. On \(|z_h|=D\) the next segment points strictly outward already at its start: the scalar product of \(z_h\) with \(z_{h+1}-z_h\) is positive by the preceding estimate. Thus crossings persist under perturbation and no discontinuity occurs when a first crossing moves across a discrete layer. At a terminal crossing the crossing and terminal-direction definitions agree. Parameter continuity is Lemma 41. On the boundary of the initial ball the estimate \(|\lambda_N-x|\le C_L\) makes this map homotopic to the radial identity. A map of nonzero boundary degree cannot extend over the ball, a contradiction. A zero terminal shot therefore exists and never exits its barrier. The two estimates above prove (105). ◻

Proof. We first check the reflection input to the lemma “Exponential moments of selected rows” in (OpenAI 2026c, Lemma 7.3 and Equation (7.5), Section “Integrated comparison of endpoint densities”). At the fine scale, cut the flat reference into two half cylinders and choose lifts in each half. Their transverse integer windings must agree. Along either separating seam, fix an origin and express each boundary site’s sheet by cumulative tangential integers. Flatness then says that every cross-link integer is the difference of these boundary sheet displacements plus one common integer. The common integers on the two seams can be summed independently.

In the lifted variables, the kernel for one seam is consequently a product Gaussian kernel and Wilson \(q\)-kernel, summed over a common deck shift. Each unsummed kernel is positive semidefinite: the Gaussian is positive definite, and \(e^{\beta q\cdot Q}\) is positive definite by its power series. The common-shift sum preserves this property. Indeed, let \(g\) denote simultaneous translation by one sheet, including its sign on \(q\), and let \(K\) be the unsummed kernel. For a finite set of boundary configurations \(z_i\), the matrices \[\frac1{2M+1}\sum_{a,b=-M}^{M} K(g^az_i,g^bz_j)\] are Gram matrices. Simultaneous-shift invariance and Gaussian decay show that their limit is \(\sum_{n\in\mathbb Z}K(z_i,g^nz_j)\). The restrictions on the flat halves and their equal transverse winding are diagonal restrictions on these positive forms. Tangential directions are chosen the same in the two halves, and normal integers transform with their orientation. This proves seam reflection positivity of the untwisted flat reference.

A minus-center twist inserts the isometry \(q\mapsto-q\) in the seam kernel. Cauchy–Schwarz therefore bounds a twisted pairing by the geometric mean of its two untwisted doubled-half pairings. The twist can be moved to any desired seam by a change of representatives. These statements pass through the exact observations. A half-observable uses only internal output sites and integers; its fine data and observation noises lie in that same half. The observation kernels are reflection equivariant on the lifts. Cross-link output integers are not read by such an observable and sum out by normalization. Thus the fine reflections can be taken at the block seams corresponding to the terminal reflections.

We can now use the chessboard argument. In a translated grid of \(B_T\)-squares, retain the rows in \(I\) whose complete stencils are internal to one tile, with a margin \(C_LD_1\). The tile test is the exponential of their sum. Reflection disseminates a tile test to a subset of the kinetic rows without repetitions. To keep the integral over the unbounded lift finite, use coefficient \(1-\delta\) in this test, where \(\delta>0\) is a prescribed inverse power of \(\log H\). The unspent fraction \(\delta\) of every direct term supplies spanning tree integration and all deck sums. With (109), the absolute covering cap, and (108), a fully disseminated test has normalized integral at most \(e^{C_LV\log H}\). Both its numerator bound and its denominator lower bound have the same translation factor \(P\).

Taking the chessboard root gives a cost \(e^{C_LB_T^2\log H}\) per occupied tile, and there are at most \(|I|\) occupied tiles. For a time twist, first place it at a tile seam and apply the preceding Cauchy–Schwarz bound. Both doubled arrays are untwisted, so their geometric mean has the same tile-counting cost and the same denominator \(Z_{F,+}^{\rm end}\).

Finally average over the \(B_T^2\) translations of the grid. Each row is internal for a fraction at least \(1-C_LD_1/B_T\) of them. Generalized Hölder and \(Y_e\ge0\) give (110) with \[\vartheta=(1-\delta)(1-C_LD_1/B_T).\] Reflection covariance permits the twist to be moved separately for each grid. Choose \(\delta\) small and then \(B_T\) large to obtain the stated accuracy. Truncating the nonnegative tests first and then using monotone convergence justifies the argument for unbounded rows. ◻

Proof. At every lift of a point of \(U^c\), all squares containing that point have average gradient at most \(\alpha\). The concentric-average proof in the cited lemma applies on the whole cover: the discrete \(L^1\) Poincaré inequality compares the point with averages on doubled squares, at scale \(R\) at cost \(CR\) times the average gradient. Compare two such averages at a scale comparable to their separation and sum the geometric scales. This gives, in the representatives transported by \(k\), \[|q_x-q_y|+|w_x-w_y|\le C\alpha|x-y|_\infty \qquad(x,y\text{ above }U^c).\] All sizes were admitted, so this estimate does not stop at the shorter period of the rectangle.

Extend the real coordinate by distance cones. For example, the infimum of \(w_y+C\alpha|x-y|_\infty\) over prescribed lifts \(y\) is finite, agrees with the prescribed data, and is Lipschitz. The set of prescribed lifts includes every deck translate, so the extension has exactly the same additive winding under translation by a period.

For \(q\), first extend its four Euclidean coordinates near the prescribed data. Averaging the upper and lower distance-cone extensions, and clipping to the bounded coordinate range if needed, makes this construction equivariant under the required sign changes. Within distance \(c/\alpha\) of the prescribed set its norm is bounded away from zero; normalization gives an \(S^3\)-valued extension with Lipschitz constant \(C\alpha\). Choose a rectangular mesh of spacing comparable to \(c'/\alpha\), with \(c'\) sufficiently smaller than \(c\), and preserve that extension in each mesh box meeting prescribed data. Choose the remaining vertices and shortest edge arcs per mesh orbit, then extend by the prescribed signs to all translates. Each unfilled box has a boundary map with a fixed Lipschitz bound after rescaling.

Such a loop has a uniformly Lipschitz filling in \(S^3\). One concrete construction chooses a point separated from the antipodal image of the loop and normalizes the cone from that point to the loop. The antipodal curve has bounded length, so a fixed-radius tubular covering occupies less than the volume of \(S^3\); a point outside it exists. Ordered finite nets and fixed measurable choices of shortest arcs give measurable fillings. This is the quantitative filling construction of (OpenAI 2026b, Lemma “Quantitative filling on \(S^3\),” Section “Relative comparison of the retained densities”). Adjacent fillings use the same previously fixed edge maps. Defining choices by mesh orbits also makes the resulting map sign-equivariant on the cover. Both periods exceed the mesh size, which is the only period-size requirement in this construction.

Sample the replaced spins independently in balls of radius \(c't_0\) around these extensions, using Haar measure in the sphere factor and Lebesgue measure in the real factor. Extend the samples to all translates with the original winding, and reduce to the chosen representatives, inducing the edge integers and the corresponding signs on \(q\). The extension equals the original data outside \(U\). Taking \(\varepsilon\) and \(c'\) sufficiently small therefore makes every new internal or boundary chord smaller than \(t_0\). The collar of each \(E_C\) ensures that no originally flagged bond crosses its boundary. Any subset of the regions can consequently be replaced independently, with exactly its inventories removed. Exterior and unselected data keep their original representatives and integers.

If there are no prescribed data, perform the same mesh construction in the trivial integer winding, with the fixed time sign still imposed. For a whole-torus replacement one may use this construction and add a uniform common translation to the real coordinate. Each local sampling ball has four-dimensional volume bounded below by a fixed multiple of \(t_0^4\). Hence the fill density is at most \(e^{C_Ln\log H}\). In the whole-torus case the uniform translation supplies the additional factor \(1/P\), with any remaining fixed or polynomial factor in \(H\) included in the same exponential bound.

It remains to check the energy estimates without an aspect-ratio loss. The image of a covering square may wrap around either period. In a wrapped coordinate the numbers of preimages of its sites differ by at most a factor two; thus the multiplicities in a square’s torus image differ by at most a fixed factor. A marked square therefore still provides an average-gradient lower bound, with a decreased constant, on its image. Fixed dilations of these images have uniformly bounded volume ratios. Choose finitely many generating squares, possible because the torus has finitely many sites, and select disjoint images in decreasing order of size. Fixed enlargements of the selected images cover the generators. Dilation by \(d\) costs at most \(Cd^2\), so their total area in a region is at least \(c|E_C|/d^2\).

On each selected image, either increments greater than \(t_0\) provide a fixed fraction of the average gradient, in which case the linear branch of the direct energy gives the required lower bound, or the quadratic branch does so by Cauchy–Schwarz. In both cases the energy is at least \(cH\alpha^2\) times its area. Bonds in the selected images have bounded overlap, and \(H\alpha^2=\varepsilon^2p_0^2\). This proves the first bound in (111).

Take \(I_E\) to contain all rows within distance \(C'D_1\) of \(E\). Choose the multiplier in \(d\) after the stencil constants, so that the total stencil enlargement is less than \(d/2\). It then reads no marked part of an unselected dilated component: such marked parts remain at distance at least \(d\) from \(E\). On the other unchanged sites the exterior estimate applies, and on replaced sites the sampled chords are below \(t_0\). Every row read by \(I_E\) is therefore bounded by \(C_LHt_0^2=C_Lp_0^2\). The count \(|I_E|\le Cd^2n\) proves the second bound. ◻

Proof. Only rows in \(I_E\) change. Equations (109) and (111) give the exponent comparison \[X_F^\eta(z)-X_F^\eta(z') \le C_Ln-cp_0^2n/d^2+\sum_{e\in I_E}Y_e(z').\] Retain a fixed fraction of the original direct energy on edges incident on \(E\), decreasing \(c\). The first inequality in (112) then permits replacement of the last coefficient by \(\vartheta\), at a cost of at most a fixed fraction of the negative term.

Apply the uniform bound on the conditional fill density, and integrate the original variables in \(E\) against the reserved direct weight. When \(E\ne\Lambda\), choose a spanning forest anchored in the unchanged exterior. Each tree edge integrates a real increment, while each remaining incident edge has an exponentially convergent integer sum. The spacing \(P\) is bounded below, so these costs are at most \(e^{C_Ln\log H}\), independently of \(P\). For \(E=\Lambda\), one tree root remains free and the cost is \(Pe^{C_Ln\log H}\). It cancels the factor \(1/P\) in the whole-torus fill density. Unchanged exterior data include their edge integers; only integers incident on \(E\) are summed in this operation.

The final factor \(\Xi_F^\eta(z')\) is a complete positive flat-reference sum. We may therefore discard the restrictions defining \(\mathcal Q\) and integrate the final configuration against the complete positive reference density, retaining its flatness constraint. The resulting upper bound is \[\exp\{C_Ln\log H-c''p_0^2n/d^2\} \frac1{Z_{F,+}^{\rm end}} \int_{\mathcal F}e^{X_F^\eta}\Xi_F^\eta e^{\vartheta\sum_{I_E}Y_e}\,d\nu.\] There is no injective-change-of-variables assertion in this argument: the original and filled variables are integrated separately using the conditional-density bound. Apply (110), \(|I_E|\le Cd^2n\), and the second inequality in (112) to obtain (113). ◻

Proof of Proposition 46. We use the identities and forest estimate of (OpenAI 2026c, secs. 7.4–7.5). The relevant subsections are “Exact covering identities and integration” and “Summing the forests.”

Fix the original regions and their inventories, and sample their fillings once. If \(I\) is a set of regions left unfilled, write \(z(I)\) for the resulting configuration. Then \[D(z(I))=\bigsqcup_{C\in I}D_C.\] Write \(\Xi(I)=\Xi_F^\eta(z(I))\) and \(\widetilde\Xi(I)=\widetilde\Xi^\eta(z(I))\). No filled configuration is reclustered.

In a complete covering of these active inventories, join labels whenever their supports meet the same active region. Each connected group is a macrolabel: its enlarged support consists of its label supports and all active regions they meet, and its weight is the product of their weights. It covers in full every active region it touches. Distinct macrolabels have disjoint enlarged supports. For a site set \(S\) disjoint from the active regions, let \(\Xi(I;S)\) be the complete-inventory reference sum restricted to macrolabels avoiding \(S\). This restricted sum may be signed. Write \(w_U(z(I))\) for a macrolabel’s product of reference weights, and \(\widetilde w_U(z(I))\) for the corresponding product of comparison weights; their enlarged supports are denoted by \(S_U\).

Selecting a compatible macrolabel family \(\mathcal A\), with combined support \(S_{\mathcal A}\) and consumed regions \(B_{\mathcal A}\), leaves the background sum \(\Xi(I\setminus B_{\mathcal A};S_{\mathcal A})\). Each selected macrolabel consumes every active region its support meets, so \(S_{\mathcal A}\) is disjoint from the remaining active regions, as required in the definition of this restricted sum. Every remaining label avoids all changed sites and incident integer data, so its value and all eligibility tests are unchanged. Its inventory loses exactly the consumed inventories. Expanding the avoidance indicators by inclusion–exclusion, and expanding the difference of the two full products, gives \[\begin{align*} \Xi(I;S) &=\sum_{\substack{\mathcal A\ \mathrm{compatible}\\ S_U\cap S\ne\varnothing\ (U\in\mathcal A)}} (-1)^{|\mathcal A|} \prod_{U\in\mathcal A}w_U(z(I))\, \Xi(I\setminus B_{\mathcal A};S_{\mathcal A}),\\ \widetilde\Xi(I)-\Xi(I) &=\sum_{\substack{\mathcal A\ \mathrm{compatible}\\ \mathcal A\ne\varnothing}} \prod_{U\in\mathcal A} [\widetilde w_U(z(I))-w_U(z(I))]\, \Xi(I\setminus B_{\mathcal A};S_{\mathcal A}). \end{align*}\] All macrolabels in these sums satisfy the full-inventory conditions just described. These are the cited exact identities. Iterate the first in the background factors arising from the latter. Its empty-family term is a complete reference sum, while every nonempty generation consumes an active region. Thus the recursion terminates at complete reference sums \(\Xi_F^\eta(z')\).

Take absolute values only after these identities. Telescoping a macrolabel product difference marks one of its original labels, at cost \(d_\ell\), while all other label factors cost \(b_\ell\). The deterministic forest encoding from the cited proof has original labels and exact region site sets as vertices: internal edges describe each macrolabel, and a second edge type joins a later macrolabel to an intersecting one in the preceding generation. Root each component at its marked label. No region is repeated, because it is consumed at first use. No original label is repeated, because its fixed nonempty inventory becomes ineligible once consumed. The forest retains the generations, the marked choices, and every evaluation configuration. Thus the same counting applies with the present incident-integer convention. The common label list includes labels whose evaluated weight is zero; all spin-dependent eligibility tests stay in the evaluated weights. For a fixed encoded term the set \(\mathcal Q\) in Lemma 45 is consequently defined from the original geometry and inventories before the fillings are sampled.

Lemma 45 assigns a region of \(n\) sites weight \(e^{-c'p_0^2n/d^2}\). On the rectangular torus the number of connected site animals of size \(n\) through a given site is at most \(C^n\), with a constant independent of either period. Let \(R=C_LM_0^2\) be a fixed support-to-load bound, so that \(|P_\ell|\le Rs_\ell\). For \(H\) large, \[\sup_x\sum_{E\ni x}e^{-c'p_0^2|E|/d^2}e^{2|E|} <\frac1{16R},\qquad \sup_x\sum_{\ell:x\in P_\ell}b_\ell e^{2s_\ell} <\frac1{16R}.\] Summing over an intersection site and the two edge types bounds the one-child sum at a vertex of size \(s\) by \(s/4\), including an exponential allowance for descendants. Induction on tree depth bounds all descendants by \(e^s\). A marked root consequently costs at most \(d_\ell e^{2s_\ell}\). Dropping disjointness between the resulting rooted trees and summing unordered nonempty root families gives \[\sum_{j\ge1}\frac1{j!} \left(\sum_\ell d_\ell e^{2s_\ell}\right)^j,\] which is (114). The support-hit bounds justify passing from finite to countable label lists. Positivity has been used only for complete flat-reference sums at the filled configurations, never for a restricted signed sum. A common multiplier of modulus at most one can be removed in this absolute estimate, proving the last assertion as well. ◻

Completion of Proposition 28. Apply Proposition 46 with the Brownian covering sum on \(\mathcal F\). By (106), the sum in (114) is at most \(C_HV e^{-c_H\beta}\), and the scalar-stripped exponents have the same supremum discrepancy. The direct, unbounded rows agree exactly at the matched terminal couplings; all remaining exponent differences are bounded. The integrated estimate therefore also permits replacing the Brownian exponent by the reference exponent. Since \(V\) is polynomial in \(\beta\), this gives an error \(O(e^{-c\beta})Z_{F,+}^{\rm end}\), after decreasing \(c\). Add (107).

Finally insert the Brownian terminal phase in both flat integrals. Its modulus is one, with no bound required on its coefficient. On the reference side exact transport of winding through the observations identifies it with the initial multiplier \(\exp(i\phi\sum d_tv)\), for the corresponding real \(\phi\). Reassemble the exact observations and remove each regulator’s accumulated bulk scalar. The two scalar coefficients need not agree, but each is independent of the periods and the time seam. The same coefficient for each regulator is therefore used for both seams and for time lengths \(m_t\) and \(2m_t\). The resulting scalar-stripped partition functions satisfy (76), proving Proposition 28. ◻

Proof. We first obtain the lower bound at the matched parameters, including the phase in (76). In the initial variables, the flat reference is the product of a Wilson integral and a Gaussian lift, summed over their compatible windings. If \((k_x,k_t)\in\mathbb Z^2\) is the winding pair, the harmonic part of the Gaussian energy is \[\frac{\pi^2}{2} \left(c_x\frac{m_t}{m_x}k_x^2+ c_t\frac{m_x}{m_t}k_t^2\right), \qquad c_x,c_t=\beta+O_L(1).\] The Gaussian fluctuation integral is the same in every winding sector. The corresponding Wilson integral has the center twists prescribed by \((k_x,k_t)\), and is at most its untwisted value. This last bound follows by expanding the Wilson bonds in powers: after spin integration the coefficients with no seam signs are nonnegative, and adding a center seam only inserts signs.

Let \(A_0\) denote the positive zero-winding contribution to the untwisted reference partition function. Since the phase has modulus one and is identically one at zero winding, the total absolute contribution of all other windings is at most \(\varepsilon_\beta A_0\), where \[\varepsilon_\beta \le \sum_{(k_x,k_t)\ne(0,0)} \exp\!\left\{-c\beta\left( \frac{m_t}{m_x}k_x^2+ \frac{m_x}{m_t}k_t^2\right)\right\}=o(1).\] Indeed, \(\beta\min(m_t/m_x,m_x/m_t)\to\infty\) for the chosen rectangles. Pairing opposite windings makes the phase-weighted reference integral real. It is therefore at least \((1-\varepsilon_\beta)A_0\), whereas the reference integral with phase zero is at most \((1+\varepsilon_\beta)A_0\). These comparisons are unchanged when the common reference volume scalar is removed. In particular, \[Z^{F(\phi)}_+\ge \frac{1-\varepsilon_\beta}{1+\varepsilon_\beta}Z^{F(0)}_+.\]

Consequently the phase-weighted reference retains, up to a fixed factor, the aligned-cap lower bound established above. The integrated comparison (76), obtained from (105), transfers this lower bound to the matched Brownian integral after its volume scalars are removed. At the terminal layer that integral is thus at least \[cP\exp(-C_L V\log H),\] where \(P=\pi R_0\) is the terminal lift period.

For the upper bound at \((b',r')\) we need only a short initial part of the renormalization. Initialize both runs with the same fixed \(R_N\), and perform \(s=\lceil\log b/\log L\rceil=O_L(\log b)\) steps; in particular, \(s\ll N\). At forward step \(h\le s\) the depth index is \(j=N-h\) and \(H_j\asymp b\). The initial displacement in the displayed couplings is bounded, and each coefficient step, with the prescribed coordinate rescalings, costs at most \(C_L\). The displacement and the accumulated corrections are therefore at most \(C_L(1+h)\). The admission bands have room for this initial segment at both parameter values.

After \(s\) steps the layer contains \(L^{-2s}V_{\mathrm{site}}\) sites. The spanning-tree integration bound and the class caps, with the phase replaced by its absolute value, bound the remaining integral by \[P_{N-s}\exp\!\left( C_L L^{-2s}V_{\mathrm{site}}\log b\right).\] Our choice of \(s\) gives \(L^{-2s}\log b\le C_L/b\). The logarithm of this integral then costs \(O_L(1/b)\) per initial site. The single factor \(P_{N-s}\) has the same property: \(|\log P_j|=O_L(\log b)\) throughout the trajectory, and \(V_{\mathrm{site}}=L^{2N}V\) grows exponentially in \(\beta\).

It remains to compare the extracted scalars on this short segment. Let \(S_s(b,r)\) denote their sum per initial site, including initialization and the first \(s\) steps. A scalar extracted per output site at forward step \(h\) has weight \(L^{-2(h+1)}\) in this sum. By (104), the initial scalar difference is \[b'r'-br+O_L(1/b).\] Here the changes in \(\log g\) and in the bounded Lipschitz initialization term are \(O_L(1/b)\), and the two scalar remainders have that size as well. On the subsequent short segment, the bounds \(H_j\asymp b\) and \(C_L(1+h)\) on coefficient drift show that the shape, \(\log g\), and \(\chi\) arguments in (89) differ by at most \(C_L(1+h)/b\). The rescaling schedule and the type of free history are the same in the two Brownian runs. The Lipschitz bound for the displayed terms in (89), together with the absolute \(O_L(1/H_j)\) bounds on its two remainders, consequently gives \[\begin{align*} S_s(b',r')-S_s(b,r) &=b'r'-br+ O_L\!\left(\frac1b\right) \left(1+\sum_{h=0}^{s-1}(1+h)L^{-2(h+1)}\right)\\ &=b'r'-br+O_L(1/b). \end{align*}\]

For the matched run, the scalars extracted after these \(s\) steps also have total absolute size \(O_L(1/b)\) per initial site. Indeed (89) bounds each by \(C_L\log b\) per output site, and their remaining volume weights sum to \(O_L(L^{-2s})\). The terminal lower bound above has logarithm bounded below by \(\log(cP)-C_LV\log H\), whose negative part, divided by \(V_{\mathrm{site}}\), is again \(O_L(1/b)\). We have therefore proved \[\frac{\log Z^B(b',r')}{V_{\mathrm{site}}} \le S_s(b',r')+O_L(1/b),\qquad \frac{\log Z^B(b,r)}{V_{\mathrm{site}}} \ge S_s(b,r)-O_L(1/b).\] Subtracting and using the scalar comparison proves (115). The argument uses the full matched trajectory only for the lower bound; the displaced parameters require just the admitted short segment. ◻

Proof. We use the finite-volume coefficient method of (OpenAI 2026b, Section “Determining the kinetic drift,” subsection “Computing the same coefficients after blocking”). Its inputs here are the finite Taylor maps established in (86), the Brownian initialization (101)–(102), and the matched coefficient bounds (105). The partition comparison (76), which concerns center seams on long rectangles, is not being differentiated.

Fix \(m,s\) and put \(n=mL^s\). For sufficiently large \(\beta\), the trajectory has at least \(s\) layers. After \(s\) observation steps the period is \(m\), and the depth index of the original trajectory is \(N-s\). We will prove \[ h_i^M(mL^s) =B_{i,N-s}^{q,M}+\mathcal C_i^M(m,s;\beta) +O(C_Lm^D e^{-cm})+o_{\beta;m,s}(1), \qquad M\in\{F,B\}. \tag{117}\] Here \(\mathcal C_i^M(m,s;\beta)\) is the Gaussian correction from the normalized order-two logarithmic expansion at period \(m\), computed from the current canonical coefficients, shapes, and full free kernel. It includes the contribution of two order-one vertices. Its dependence on \(\beta\) is through those current data; we do not assume that they converge as \(\beta\to\infty\). The leading coupling must be identified exactly. The constant-order correction and the displayed winding error will then be controlled uniformly in \(s\), whereas the last remainder needs only tend to zero for fixed \(m,s\).

Insert \(s\) normalized observation kernels at the actual parameters of the matched trajectories. Choose the observation blocks inside the fundamental cell. Near an aligned configuration use the same equivariant formulas as in the renormalization construction; away from this neighborhood complete them by any local equivariant probability kernels. These completions need not depend on \(\zeta\). For \(B\), observe in the block lift near alignment and then reduce to the quotient. For \(F\), first omit nonzero circle winding and work in its real lift. At fixed \(n\), these omitted sectors and their first two quaternion-twist derivatives have relative size \(O_n(\beta^C e^{-c_n\beta})\): harmonic circle winding has a positive cost of order \(\beta\), and differentiating the finitely many twisted bonds costs only a polynomial in \(\beta\). Thus their contribution to the helicity is \(o(1)\). The normalized observation insertions leave the retained partition functions unchanged.

Right twisting preserves the common left action on the quaternion field, so one may pin a last-layer quaternion spin. The common circle translation may also be pinned. More explicitly, subtract one output circle coordinate from every lifted coordinate. Integration of the remaining common translation over one period contributes exactly that period, independently of the other coordinates. This remains true if it was originally one microscopic circle value that was required to lie in a fundamental domain. The extracted phase is identically one on zero-winding configurations, since \(\sum d_t v=0\); it therefore contributes no helicity term.

We next justify the fixed-volume expansion and its identification with the bulk Taylor maps. After the two symmetries have been removed, the aligned saddle at \(\zeta=0\) is unique and its transverse quadratic form is strictly positive. Count the angle \(\zeta\) at the same scale as the chart fields, namely \(g\asymp\beta^{-1/2}\). At fixed \(n,s\) the Wilson expansion is an ordinary Laplace expansion. For the Brownian expansion, the endpoint and path estimates used in (101)–(102) give the same conclusion. Indeed, restrict endpoint bond distances to \(g\) times a fixed power of \(\log(g^{-1})\), and restrict the path-size variables of the initialization to a corresponding logarithmic bound. The endpoint and bridge tail estimates, together with the vertical endpoint-bond energy bound, make the complement smaller than any prescribed power of \(g\). They remain valid after two derivatives in the scaled twist, by the endpoint-translation estimate. Integer lifts in the horizontal heat kernel are summed using its Gaussian bounds. Observation kernels similarly localize their variables in the aligned charts. Thus the relevant Taylor coefficients are Gaussian polynomial moments. Writing \(\eta=\zeta/g\), expand the logarithm through normalized order two. At fixed \(n,s\) its remainder and its first two \(\eta\) derivatives at zero are \(o(g^2)\), uniformly for the actual shape and coordinate-scale parameters in their compact bands. Since \[h_i^M(n)=-g^{-2}\left.\partial_\eta^2 \log Z_i^M(n;g\eta)\right|_{\eta=0},\] this remainder contributes \(o(1)\) to the unscaled helicity. For fixed \(s\), all observation widths are comparable to \(g\).

To identify these coefficients, complete the Gaussians using full kernels rather than their numerical truncations. Extend each background field across the boundary with the prescribed right twist. Ambient linear kernels commute with this continuation, because right multiplication is orthogonal on \(\mathbb R^4\); their compositions and adjoints fold with the same transport. If \(T_x^q\) is the coarse reference field and \(q_x=\exp(\xi_x)T_x^q\), then \(\xi_x\) is ordinary periodic. In fact both \(q_x\) and \(T_x^q\) acquire the same right multiplier on crossing a period, which cancels in \(q_x(T_x^q)^{-1}\). The affine row in the chart expansion therefore has the usual folded matrix on \(\xi\) and a background term that is also ordinary periodic. The leading conditional Gaussian precision in the joint expansion, with both the twist and chart fields counted at scale \(g\), is therefore the ordinary folded precision. Higher-order background and twist dependence remains among the Taylor vertices. This argument also applies to horizontal seam strips in (101)–(102): endpoint transport leaves the left tangent logarithms and bridge variables periodic.

Consequently every Wick contraction without a winding obstruction has the bulk formula, evaluated on the fields lifted along its contraction paths. Removing the anchored quaternion rotation cancels the right transports. The higher-order determinant and Jacobian terms remain among the Taylor vertices. The logarithmic determinant coefficients are sums over closed contraction paths and obey the same identification. Repeated transports on a path contribute at most a fixed power of its length to the twist jets under consideration, so the exponential path norms still sum absolutely. Linear circle terms telescope in this saddle calculation.

There is one leading-order point for which an exponentially small estimate would be insufficient. At harmonic order the first jet of a twisted field is an affine slope plus an ordinary periodic field. The full free kernels preserve affine fields. For every folded free edge matrix, translation invariance makes the cross term between a constant slope and periodic gradients exactly zero. One can see this also at the last layer by using a linear ramp that vanishes at the pinned site: its sum with an ordinary periodic field describes every first jet with the given seam increment. The affine Hessian normalization therefore gives, exactly, the current directional coupling \(B_i^q\) as the leading unscaled helicity on the square. The leading determinant is independent of the angle. The strip kernels have this same property by (78). In particular no term of the form \(\beta\) times an exponentially small winding error is discarded. The coupling resets in (86), including those at initialization, allow us to use the actual running couplings: at fixed \(s\) the remaining coefficient error is \(o(1)\) and multiplies a chart quadratic.

It remains to control the constant-order terms uniformly as the number of formal steps grows. At an output period \(m'=mL^j\), a discrepancy between a finite-period Wick coefficient and its folded bulk value requires contraction paths to travel a distance at least \(cm'\). The path estimates proved for both initializations give a bound \(C_Le^{-cm'}\) per anchor; twist derivatives introduce only fixed path powers. A previously generated discrepancy grows by at most \(C_L\) in one subsequent formal step. This may be estimated in the total absolute polynomial coefficient norm, including the number of anchors at the layer where it was generated: prediction and Wick substitutions have bounded row moments, and a new vertex attached to an old discrepancy is summed with the same intervertex covariance bound. The total angle is another field variable in this calculation. Its occurrences in predictions cost path moments rather than the period. Products of discrepancies satisfy the same polynomial estimates.

Thus all winding contributions through normalized order two, including changes of coupling convention, are bounded by a fixed polynomial times \[\sum_{j\ge0}C_L^{j+1}e^{-cmL^j}.\] At the final period the pinned Gaussian moments grow at most polynomially in \(m\), by uniform ellipticity. Enlarging a fixed exponent \(D\) gives the bound \(C_Lm^D e^{-cm}\) for their contribution to the constant-order helicity, uniformly in \(s\) after the inner limit. Normalized order-one polynomials have odd total degree in the scaled angle and Gaussian fields, and hence have zero direct second derivative after Gaussian integration. Their propagation into order two is already included in the bound. The estimate includes winding paths whose contractions leave a function of only a few sites.

This proves (117): earlier bulk scalar terms have no angle dependence, and the remaining final-period Gaussian correction is \(\mathcal C_i^M(m,s;\beta)\). After separating the leading coupling, the normalized coefficient slots have fixed caps and the shapes lie in compact bands. At fixed \(m\), pinning removes the kernel of the uniformly elliptic free precision. The Gaussian moments and convergent path sums defining the order-two logarithmic correction are therefore Lipschitz in these data, with a constant depending on \(m\) but independent of \(s\).

By (105), the directional coupling discrepancies and canonical coefficient discrepancies at this layer are \(O(\rho^s)\). The free-history estimates add a term tending to zero with \(s\). Subtract (117) for \(F\) and \(B\), first let \(\beta\to\infty\) at fixed \(m,s\), and then let \(s\to\infty\). The result is bounded by \(C_Lm^D e^{-cm}\), which tends to zero with \(m\). This proves (116). The use of fixed-volume expansions is legitimate even when \(n\) is smaller than a numerical cutoff required for the exact retained-density representation: only the full-kernel formal coefficients are used in this argument. ◻

Proof. Distribute the twist uniformly over the links in its direction. We may use left multiplication, by the inversion symmetry noted above. In the Wilson model write the imaginary part of a spin, relative to its aligned value, as \(\beta^{-1/2}\pi\), where \(\pi\) has three real components. The leading Gaussian covariance of each component is \(D^{-1}\). The second derivative of the bond factor contributes its scalar product, whose expectation gives \[\beta-\frac{3}{2}\int_{W,n}\frac{D_i}{D}.\] The first derivative contains the chosen component of \(\operatorname{Im}(q_{x+e_i}q_x^{-1})\). Its linear term telescopes on summing over links. Its quadratic term, up to an irrelevant sign, is \(\pi_x\times\pi_{x+e_i}\). Choosing the first current component, put \[S_i=\sum_x \bigl(\pi_x^2\pi_{x+e_i}^3-\pi_x^3\pi_{x+e_i}^2\bigr) =\sum_x\pi_x^2(\pi_{x+e_i}^3-\pi_{x-e_i}^3).\] Wick contraction yields \[n^{-2}\mathbb E S_i^2 =4\int_{W,n}\frac{\sin^2 k_i}{D^2}.\] Subtracting this current variance proves (118). Every quantity in this calculation is unchanged by adding a common field constant. Thus pinning one spin or using the mean-zero propagator gives the same answer. Nonzero circle windings in \(F\) are exponentially small at fixed \(n\) and do not alter this formula.

For \(B\), the Brownian quadratic energy and (102) give four independent tangent fields at scale \(b^{-1/2}\): three quaternion components \(\pi\) and the circle component, denoted \(\nu\). At \(r=1\) their covariance is \(D_*^{-1}\). The vertical scalar product is multiplied by \(\exp(i\Delta_t v)\), so all four fields contribute to its quadratic correction. This gives \(br-2\int_{B,n}D_t/D_*\). The linear circle term again telescopes. The quadratic vertical current now has the form \(S_t\pm iT_t\), where the horizontal coordinate is continuous: \[\begin{aligned} S_t&=\sum_{u=0}^{n-1}\int_0^n \bigl(\pi^2(x,u)\pi^3(x,u+1) -\pi^3(x,u)\pi^2(x,u+1)\bigr)\,dx,\\ T_t&=\sum_{u=0}^{n-1}\int_0^n (\Delta_t\nu)(x,u)(\Delta_t\pi^1)(x,u)\,dx. \end{aligned}\] The components occurring in \(S_t\) and \(T_t\) are independent. Hence \[n^{-2}\mathbb E S_tT_t=0,\qquad n^{-2}\mathbb E T_t^2=\int_{B,n}\frac{D_t^2}{D_*^2}.\] The derivative of the logarithm of a complex weight uses the ordinary bilinear second moment, with no complex conjugation. Thus the \(iT_t\) term contributes the negative of this last quantity to the current variance. Together with the Wilson current calculation, this is exactly the vertical formula in (119).

For the horizontal response, rotate each Brownian path steadily on the left. The deterministic-rotation identity for the heat measure used in (101), in Stratonovich convention, gives the constant quadratic stiffness \(b/r\). Its first derivative is the integrated current \(\int\operatorname{Im}(dq\,q^{-1})\) in direction \(e\), multiplied by \(b/r\) and by the angle-per-period factor. The linear tangent term telescopes. The remaining Gaussian current is \[\sum_t\int_0^n(\pi^2\,d\pi^3-\pi^3\,d\pi^2),\] whose variance, divided by \(n^2\), is \(4\int_{B,n}k_x^2/D_*^2\).

For completeness, this last identity can be obtained without treating a Brownian derivative as a pointwise field. Interpolate the paths on a dyadic polygonal mesh of spacing \(\ell=n/N'\), including all strip endpoints, and let \(N'\to\infty\). The leading fields consist of independent Gaussian bridges with the deterministic interpolation and Gaussian \(C^1\) additions from the strip completion. Polygonal Stratonovich approximation therefore converges with every fixed moment; there are no jumps at the finitely many joins. The point covariance has an absolutely summable horizontal Fourier series. Sampling replaces its coefficient at \(-N'/2<j\le N'/2\) by the sum of coefficients at \(j+uN'\), \(u\in\mathbb Z\), with the zero mode still omitted. The antisymmetric polygonal area has Wick multiplier \[\frac{4\sin^2(2\pi j/N')}{\ell^2}.\] At fixed \(n\), the added covariance aliases are \(O_n(N'^{-2})\) uniformly in \(j\). The multiplier is \(O_n(j^2)\) and the original covariance is \(O_n((1+|j|)^{-2})\), so summing the alias errors gives a quantity tending to zero. The remaining sums converge to the displayed Fourier variance. Subtracting a constant for pinning does not change the area. This proves the horizontal formula. Finally, since \(r\to1\), continuity of each fixed-\(n\) correction permits replacing \(r\) by \(1\) there, while retaining \(b/r\) and \(br\) in the leading terms. ◻

Proof. Average the two directional formulas in Lemma 49 and compare them by (116). The leading Brownian average is \[\frac b2(r+r^{-1})=b+o(1),\] since \(r-1=O(\beta^{-1})\). The identities \(\sin^2 k_i=D_i-D_i^2/4\) express the difference of the constant terms as paired momentum sums. We must pair them before passing to integrals, since each propagator sum diverges as \(n\to\infty\). On their common momentum region, \[\frac1{D_*}-\frac1D =\frac{D_x-k_x^2}{D_*D}=O(1) \qquad(k\to0).\] The other resulting integrands are locally bounded, and the Brownian tails in \(k_x\) are absolutely and uniformly summable. Riemann convergence therefore applies to the paired expression.

To respect the order in (116), start with any subsequence along which the bounded difference \(\beta-b\) converges. First take its \(\beta\) limit at fixed \(m,s\), then let \(s\) and \(m\) tend to infinity as in (116). This determines that limit point from the paired integrals. Equivalently, \[ \begin{split} \beta-b={}&\frac34-\int_B\frac{D_t}{D_*} +\int_B\frac{D_t^2}{D_*^2} -\frac12\int_W\frac{D_x^2+D_t^2}{D^2}\\ &\hspace{12mm}-2\left(\int_B\frac1{D_*}-\int_W\frac1D\right)+o(1). \end{split} \tag{121}\] Here \(\int_W\) means \((2\pi)^{-2}\) times integration over \([-\pi,\pi]^2\), and \(\int_B\) means the same normalization over \(\mathbb R\times[-\pi,\pi]\). The propagator difference is taken with a common neighborhood of the origin deleted before its size tends to zero. In the finite sums the constant \(3/4\) is replaced by \(3(1-n^{-2})/4\), which has the same limit. It remains to evaluate the constants in (121).

Set \(d=2|\sin(k_t/2)|\) and use a vertical angular average denoted by \(\langle\cdot\rangle\). Horizontal integration gives \[\int_{\mathbb R}\frac{dk_x}{2\pi(k_x^2+d^2)}=\frac1{2d}, \qquad \int_{\mathbb R}\frac{dk_x}{2\pi(k_x^2+d^2)^2}=\frac1{4d^3}.\] Consequently \[\int_B\frac{D_t}{D_*}=\langle d/2\rangle=\frac2\pi, \qquad \int_B\frac{D_t^2}{D_*^2}=\langle d/4\rangle=\frac1\pi.\] By symmetry, the lattice term with coefficient \(1/2\) equals \(\int_W D_t^2/D^2\). The one-dimensional lattice propagator and its derivative yield \[\int_W\frac{D_t^2}{D^2} =\left\langle\frac{d(d^2+2)}{(d^2+4)^{3/2}}\right\rangle =\frac1\pi\int_0^1\frac{3-2u^2}{(2-u^2)^{3/2}}\,du =\frac12-\frac1{2\pi}.\] For the last equality an antiderivative is \(2\arcsin(u/\sqrt2)-u/(2\sqrt{2-u^2})\).

The propagator difference is \[\left\langle\frac1{2d}-\frac1{d\sqrt{d^2+4}}\right\rangle.\] To evaluate it, define \[I(z)=\int_0^{\pi/2} \frac{1-(1+z\sin^2 y)^{-1/2}}{\sin y}\,dy, \qquad z\ge0.\] Differentiation under the integral gives \(I'(z)=1/(2(1+z))\), and \(I(0)=0\). The required difference is therefore \(I(1)/(2\pi)=\log2/(4\pi)\). Substitution in (121) gives \[\frac34-\frac2\pi+\frac1\pi -\left(\frac12-\frac1{2\pi}\right)-\frac{\log2}{2\pi} =\frac14-\frac{1+\log2}{2\pi}.\] Every limit point is this number, proving (120). ◻

Proof. Let \(U(w)\) be the row transfer normalized by its top eigenvalue, and let \(E(w)=-a^{-1}\log\|U(w)|_{\Omega^\perp}\|\). Proposition 8 gives \[E(w)=\frac{m_{\rm lat}(\beta)}a+O(w^{-1/4}), \qquad \frac{m_{\rm lat}(\beta)}a\asymp1.\] By Lemma 5, the largest nonvacuum eigenvalue is attained in odd parity. Write \(s(u,w)=\operatorname{Tr}U(w)^{u/a}-1\). The long-rectangle estimate (30), together with (26), gives exponentially small trace tails at \(u=t,2t\), and also at a dyadic \(t_0\asymp t^\epsilon\) for any fixed sufficiently small \(\epsilon>0\). Indeed their prefactors grow at most polynomially in the present \(w,t\).

After dividing by the top eigenvalue to the power \(n\), the odd trace is at least \(e^{-tE(w)}\) and at most \[s(t_0,w)e^{-(t-t_0)E(w)}\le e^{-(t-t_0)E(w)}.\] Its logarithm is therefore \(-tE(w)+o(t)\). The normalized even trace is \(1+o(1)\), since it contains the vacuum and its remaining terms are bounded by \(s(t,w)\). These estimates, also at \(2t\), prove (124); the vacuum eigenvalue cancels in the time-doubling expression. ◻

Proof. The circle integral is the product of its zero-winding Gaussian integral and a sum of harmonic winding weights. A temporal winding has Gaussian cost at least \(c\beta w/t\) times its winding number squared; a spatial winding has cost at least \(c\beta t/w\) times its winding number squared. Each winding also inserts the corresponding center signs in the quaternion boundary conditions, and the temporal winding may carry the phase of modulus one.

We first estimate the effect of a spatial quaternion twist. Expanding all Wilson bonds in powers gives nonnegative integrated coefficients before seam signs are inserted. Consequently, for either time seam, \[\bigl|Z^q_{\varepsilon,+}-Z^q_{\varepsilon,-}\bigr| \le Z^q_{+,+}-Z^q_{+,-}.\] Applying the row-trace estimate along the other lattice axis bounds the right-hand side by \(Ce^{-cw}Z^q_{+,+}\). This also bounds the change in either time-parity combination.

By (124), \(Z_1^q/Z_0^q\ge e^{-Ct}\) for a fixed \(C\) and all large \(\beta\). The total contribution of nonzero temporal winding is therefore negligible even relative to the odd trace, since \[e^{Ct-c\beta w/t}\longrightarrow0.\] For spatial winding we use the preceding twist estimate instead of requiring its Gaussian cost to dominate \(t\). The sum of its nonzero relative Gaussian weights tends to zero because \(\beta t/w\to\infty\). Replacing each spatially twisted quaternion partition by the untwisted one costs, relative to either parity trace, at most a constant times \(e^{Ct-cw}\), which also tends to zero. Summing the Gaussian winding tails proves (125). The same comparisons hold at \(2t\). This argument controls the odd half-difference itself, rather than subtracting two partition estimates whose errors might exceed that difference.

To prove (126), use the initial circle variable of period \(\ell_v=\pi\) and its kinetic coefficients \(c_x,c_t\asymp\beta\); their ratio is bounded above and below. Up to a scalar exponential proportional to \(nX\), the zero-winding Gaussian integral is \[Z_G(t)= \frac{\ell_v\sqrt{nX}(2\pi/c_t)^{(nX-1)/2}} {\sqrt{\det{}'(D_t+(c_x/c_t)D_x)}}.\] The prime omits the constant mode. Factor the determinant over horizontal momenta \(p_x\in(2\pi/X)\mathbb Z\) in one period. With \[2\cosh\gamma(p_x)=2+(c_x/c_t)D_x(p_x),\] the elementary product over time momenta gives \[\sqrt{\det{}'(D_t+(c_x/c_t)D_x)} =n\prod_{p_x\ne0}2\sinh\bigl(n\gamma(p_x)/2\bigr).\] In \(2\log Z_G(t)-\log Z_G(2t)\) all terms linear in \(n\) cancel. The product terms \(\log(1-e^{-n\gamma(p_x)})\) have total absolute size \(O(1+X/n)\): use \(\gamma(p_x)\ge c|p_x|\) for centered momenta and the integrability of \(|\log(1-e^{-u})|\) on \((0,\infty)\). The remaining \(n,X\) prefactor is \(\sqrt{X/n}\), so it contributes only a logarithm of the aspect ratio, together with \(O(\log\beta)\). Since \(X/n=w/t=o(t)\), these terms prove (126). ◻

Proof. Let \(x_j\) be the eigenvalues of \(A/\tau\), with multiplicity. They belong to \([0,1]\), and at least one equals \(1\). Thus \(1\le\sum_jx_j^{2n}\le\sum_jx_j^n\). Taking logarithms gives the claim. For a direct sum, finiteness of the positive total trace makes it compact and ensures that its nonzero norm is attained; the same argument applies. ◻

Proof. Change the fugacity by \(\exp(\pm1/b)\). In the notation of (115), choose \(b',r'\) so that \[(b'/b)^2=e^{\pm1/b},\qquad r'/b'=r/b.\] The heat time and the width remain fixed. Homogeneity in fugacity multiplies the sector-\(j\) transfer by \(e^{\pm j/b}\). The numerator untwisted trace in (115) is therefore at least \(\tau_l^n e^{\pm nj/b}\). On the other hand, (127)–(128) bound the total trace at the matched parameter by \(\tau_l^n e^{Ct}\) for either \(l\). Hence \[\pm\frac{nj}{b} \le Xn\bigl(b'r'-br+O(b^{-1})\bigr)+O(t).\] Since \(b'r'=br e^{\pm1/b}\), division by \(n\) gives, for the two signs, \[\begin{aligned} j/b&\le X\bigl(r+O(b^{-1})\bigr)+O(a),\\ -j/b&\le X\bigl(-r+O(b^{-1})\bigr)+O(a). \end{aligned}\] Using \(X=Wb/r\) and \(r\asymp1\), these are the required upper and lower bounds \(j/W=b^2+O(b)\). The argument uses only the pressure estimate (115) at the displaced fugacity; it requires no matched trajectory at that parameter. ◻

Proof of Theorem 1. The depth calibration gives \(a\asymp\sqrt\beta e^{-\pi\beta}\). Since \(b=\beta+O(1)\) and \(r=1+O(\beta^{-1})\), the width in (122) satisfies \[W=\frac{rw}{ab} \asymp e^{\pi b}b^{-1/2}(\log b)^2.\] Thus the width hypothesis of (38) holds, with logarithmic exponent \(2\). Lemma 54 verifies its particle-number hypothesis for every maximizing sector in both parities. We may therefore apply (38) to (128), obtaining \[m_{\rm lat}(\beta) =32\sqrt2\,b^{1/2}e^{-\pi b}(1+o(1))+o(a).\] Multiplication by \(e^{\pi\beta}/\sqrt\beta\) turns the last error into \(o(1)\), by the depth calibration. Proposition 50 gives the remaining factor: \[\begin{split} \lim_{\beta\to\infty} \frac{e^{\pi\beta}}{\sqrt\beta}m_{\rm lat}(\beta) &=32\sqrt2\, \exp\!\left(\frac\pi4-\frac{1+\log2}{2}\right)\\ &=32\exp\!\left(\frac\pi4-\frac12\right). \end{split}\] The quantity entering (124) and (128) is the full Osterwalder–Schrader transfer gap identified in Sections 2–3, including the rotation-invariant sectors. The asserted asymptotic therefore has the full-gap meaning of Theorem 1. ◻