Context and prior work.

Haldane’s prediction, formulated in 1981 and published in 1983, distinguishes integer and half-odd-integer isotropic antiferromagnetic Heisenberg chains, with a positive bulk gap for positive integer spin (Haldane 1981, 1983a, 1983b). For spin one-half, Lieb, Schultz and Mattis had already constructed excitations of the periodic chain whose energy above the ground state vanishes with its length (Lieb et al. 1961, Appendix B). The Affleck–Kennedy–Lieb–Tasaki construction gave a rigorous gapped spin-one model (Affleck et al. 1987, 1988). Its interaction contains a biquadratic term: with \(q=\mathbf S_j\cdot\mathbf S_{j+1}\), it is proportional to \(q+q^2/3\) up to an additive constant. Our Hamiltonian has the pure bilinear interaction \(q\) and fixed endpoint fields.

The endpoint condition matters even when the bulk interaction is fixed. White and Huse numerically characterized the effective spin-one-half degrees of freedom at free ends (White and Huse 1993). To estimate bulk properties, including the gap, they added physical spin-one-half endpoint sites and adjusted their couplings. Magnetic fields on the original spin-one endpoints give a different finite-volume model, whose gap requires its own boundary estimates.

Tasaki formulated precisely this boundary problem in his work on the topological character of the spin-one Heisenberg ground state (Tasaki 2025). Equation (2) of that paper is the Hamiltonian (1); Lemma 1 proves uniqueness for every \(L\ge1\) and \(h>0\). Assumption 2 asks for a positive gap uniform in all sufficiently large \(L\), at one fixed \(h>0\). Theorem 1 resolves this fixed-boundary-field Haldane gap assertion positively and establishes that assumption. Section 7 then derives Tasaki index \(-1\) for every boundary-selected subsequential local limit.

The topological viewpoint developed through several distinct descriptions. Den Nijs and Rommelse introduced nonlocal string order (Nijs and Rommelse 1989), and Kennedy and Tasaki connected it to hidden symmetry breaking (Kennedy and Tasaki 1992). Pollmann, Turner, Berg and Oshikawa described symmetry protection through matrix-product states and the entanglement spectrum (Pollmann et al. 2010); Ogata extended the bond-centered reflection index to infinite-chain states beyond the matrix-product description (Ogata 2021). Tasaki’s local-twist index (Tasaki 2018) instead uses the twist expectation studied by Nakamura and Todo (Nakamura and Todo 2002). The boundary-state theorem in (Tasaki 2025) uses uniform \(z\) rotations and site-centered reflection. These invariances imply invariance under site reflection combined with a global \(\pi\) rotation about \(z\), the symmetry studied by Fuji, Pollmann and Oshikawa (Fuji et al. 2015). Our consequence concerns Tasaki’s index in this precise symmetry setting; it does not identify the different diagnostics above.

Methodological inputs.

For frustration-free chains, Lemm and Mozgunov proved finite-size criteria involving both interior and boundary-segment gaps (Lemm and Mozgunov 2019). Here the boundary information comes from weighted spatial moments and a coupled Gibbs-purity recurrence. The finite trial vectors use the matrix-product framework of finitely correlated states (Fannes et al. 1992) and its finite-chain formulation (Pérez-García et al. 2007); they certify energy upper bounds that initialize the recurrence.

The continuous-history representation is an instance of the time-ordered Poisson expansion of Aizenman and Nachtergaele (Aizenman and Nachtergaele 1994). The particular spatial transfer operator and finite periodic estimates used here come from (OpenAI 2026), which proves a uniform gap on even periodic rings. We retain that source’s spin normalization, coupling, and energy shift, and state explicitly the transfer and finite-input results used below. The endpoint pairings and all estimates specific to the open chain are proved here.

Proof strategy.

For each inverse temperature \(b>0\), a partition function of the periodic model is a trace of a power of a compact self-adjoint spatial operator \(X_b\). For the open chain with opposite endpoint fields, the corresponding partition function is instead a boundary pairing \(\langle v_b,X_b^n v_b\rangle\), where \(n\) is the number of sites. At even \(n\), this is a positive weighted moment of the absolute spatial eigenvalues. A rotation of the right boundary vector produces the same-field Hamiltonian in (1).

The periodic estimates show that one spatial eigenvalue dominates sufficiently high even moments. For the open chain, however, the contribution of each eigenspace is weighted by its overlap with \(v_b\), and this vector changes with temperature. Periodic concentration therefore leaves a boundary-overlap problem. We solve it by contracting the residual boundary-weighted moments while tracking the physical Gibbs purity (the sum of the squared Gibbs probabilities) of the opposite-field chain. These two estimates improve together when both length and inverse temperature double, without assuming a nonzero boundary overlap at the next temperature. Once the leading spatial line dominates the boundary pairing, its rotation invariance controls the same-field expression at every odd length in a full dyadic interval. The purity defect then decreases at a rate exponential in the inverse temperature, which yields a uniform physical gap.

Sections 2 and 3 establish the transfer representation and the periodic estimates in the form needed here. Section 4 proves the boundary recurrence. Section 5 initializes and iterates it, and Section 6 proves Theorem 1 at all the required odd lengths. Section 7 applies the resulting gap to the boundary-selected infinite-volume states. Appendices 8 and 9 give complete finite certificates for the boundary initialization. All terminating decimals used in inequalities denote exact rationals.

Hamiltonians and shifted partition functions

From now on set \[ h=\frac35,\qquad a=\frac{700741}{500000},\qquad b_0=\frac{49}{4},\qquad b_j=2^jb_0\quad(j\ge0). \tag{2}\] It is convenient to label an \(n\)-site chain by \(0,\ldots,n-1\). Write \(S_j\cdot S_k=\sum_\alpha S_j^\alpha S_k^\alpha\) and define \[\begin{align*} H_n^{\rm opp}&=\sum_{j=0}^{n-2}S_j\cdot S_{j+1} -hS_0^z+hS_{n-1}^z,\tag{3}\\ H_n^{\rm s}&=\sum_{j=0}^{n-2}S_j\cdot S_{j+1} -hS_0^z-hS_{n-1}^z,\tag{4}\\ H_n^{\rm per}&=\sum_{j=0}^{n-2}S_j\cdot S_{j+1} +S_{n-1}\cdot S_0\quad(n\ge4). \tag{5}\end{align*}\] For \(n=2L+1\), \(H_n^{\rm s}\) is (1) after relabeling. Our partition functions are ordinary traces over the entire physical Hilbert space: \[ \begin{split} Z_n(b)&=\mathop{\mathrm{Tr}}e^{-b(H_n^{\rm per}+anI)},\\ Y_n(b)&=\mathop{\mathrm{Tr}}e^{-b(H_n^{\rm opp}+anI)},\qquad Y_n^{\rm s}(b)=\mathop{\mathrm{Tr}}e^{-b(H_n^{\rm s}+anI)}. \end{split} \tag{6}\] The shift is per site, also for open chains. It will cancel from all physical Gibbs weight ratios.

The spatial operator

We use the concrete spatial-transfer operator in (OpenAI 2026, Proposition 3.1). Its auxiliary space records ordered bond events in imaginary time; each physical site contributes a kernel depending on its two incident histories. Use the unitarily equivalent Cartesian spin matrices \((S^\alpha)_{rs}=-i\varepsilon_{\alpha rs}\) and set \(G_\alpha=iS^\alpha\). Then \(G_\alpha\) is real, \(\lVert G_\alpha\rVert=1\), and \[-S\cdot S=\sum_{\alpha=x,y,z}G_\alpha\otimes G_\alpha.\] A history is an ordered list \(\omega=((t_1,\alpha_1),\ldots,(t_d,\alpha_d))\) with \(0<t_1<\cdots<t_d<b\) and \(\alpha_r\in\{x,y,z\}\); the empty list is allowed. Give the disjoint union \(\Omega_b\) of these labeled simplexes Lebesgue measure on each simplex and mass one at the empty history. Its measure \(\mu_b\) has total mass \(e^{3b}\).

For two histories, merge their events in chronological order and take the trace of the resulting product of \(G\) matrices, with increasing time from left to right. Denote this real kernel by \(k_b(\omega,\eta)\). Set it to zero at nonempty time collisions, a null set, and use trace \(3\) for the two empty histories. The kernel is symmetric: exchanging the two histories leaves their merged list unchanged. In particular, symmetry does not reverse a product of noncommuting matrices. Moreover \(|k_b|\le3\). On \(\mathcal K_b=L^2(\Omega_b,\mu_b;\mathbb C)\) define \[ (X_bf)(\omega)=e^{-ab}\int_{\Omega_b}k_b(\omega,\eta) f(\eta)\,d\mu_b(\eta). \tag{7}\] It is real, self-adjoint, and Hilbert–Schmidt. The cited transfer Proposition identifies its periodic moments: \[ Z_n(b)=\mathop{\mathrm{Tr}}X_b^n\quad(n\ge4),\qquad \sum_i|\mu_i(b)|^k<\infty\quad(k\ge2), \tag{8}\] where \(\mu_i(b)\) lists its nonzero eigenvalues with multiplicity. This exact operator, with the shift in (2), is the one used for the periodic inputs in Section 3.

Rotations will let us change one endpoint field and, later, show that the leading spatial line is fixed. Let \(D_2\) consist of the identity and the three diagonal rotations by \(\pi\), let \(C\) cyclically permute the coordinate axes, and put \(\mathcal G=\langle D_2,C\rangle\). This is a group of twelve proper signed permutations. If \(ge_\alpha=\epsilon_g(\alpha) e_{\sigma_g(\alpha)}\), write \(\sigma_g\omega\) for relabeling the marks and \(s_g(\omega)=\prod_r\epsilon_g(\alpha_r)\). The operators \[ (U_gf)(\omega)=s_g(\sigma_g^{-1}\omega)f(\sigma_g^{-1}\omega) \tag{9}\] form a real unitary representation and commute with \(X_b\). This follows either directly by rotating every matrix in the kernel, or from the same Proposition. In particular, fix \(P=\mathop{\mathrm{diag}}(1,-1,-1)\), so \(Pe_z=-e_z\).

Boundary vectors

Define \(v_b\in\mathcal K_b\) by \[ v_b(\omega)= \begin{cases} (ih)^d,&\text{all $d$ marks of $\omega$ equal $z$},\\ 0,&\text{otherwise}. \end{cases} \tag{10}\] The empty-history value is one. The simplex volumes give \(\lVert v_b\rVert^2=\sum_{d\ge0}b^dh^{2d}/d!=e^{bh^2}\). The inner product below is conjugate linear in its first argument.

For even lengths, the opposite-field expression is the positive moment \(Y_n(b)=\langle v_b,|X_b|^n v_b\rangle\). At odd lengths, the powers of \(X_b\) retain the signs of its eigenvalues. We first concentrate the even opposite-field moments. The resulting leading-line estimate will then control the signed same-field pairing at odd lengths; Figure 1 distinguishes the two boundary pairings.

Starting residual and physical-purity bounds

The thermal estimates bound the residual and leading contributions to \(Y_m(b_0)\), while the variational estimate supplies a lower bound for one ground weight. Keep \(\ell,U\) from the periodic seed and define, for each of the four lengths in (41), \[ \epsilon(m)=.027e^{b_0\alpha}U^m\frac{.874^{m-6}}{\ell^6}, \qquad R(m)=e^{.132}U^m+\epsilon(m)-1. \tag{43}\]

In particular \(\epsilon(34)<.0928<1\), proving \(c_0>0\). The estimates \(\rho_1\le.953\) and \(\rho_2\le.971\) now allow two applications of Lemma 7: \[\begin{align*} e_{78}(1)&\le F\bigl(.953^{44},.093,(1+R(78))^{-2}\bigr)<.133, \tag{45}\\ e_{142}(2)&\le F\bigl(.971^{64},.133,(1+R(142)^2)^{-2}\bigr)<.14. \tag{46}\end{align*}\] The bound at \(m=240\), \(j=2\) gives \(1-A(240,2)<.21\). Thus the induction of Proposition 8 starts with \[ E_2=.14,\qquad\eta_2=.21. \tag{47}\] Both new leading overlaps have also been proved positive by the strict defect bounds; no overlap was assumed at the next temperature.

Explicit finite comparisons

We record numerical margins to make the initialization reproducible. The three lower bounds from Lemma 10 used above are \[ \begin{split} (1+R(78))^{-2}&>.6180,\\ (1+R(142)^2)^{-2}&>.7533,\\ (1+R(240)^4)^{-2}&>.7923. \end{split} \tag{48}\] Using the full expressions in (43) gives upper outputs below \(.132129\), \(.137301\), and \(.207518\) in (45), (46), and \(1-A(240,2)\) respectively. The lower bounds in (48) are rounded for display.

Starting from (47), the following rounded bounds satisfy Proposition 8: \[ \begin{array}{c|cc|c} j&E_j&\eta_j&\text{upper bound on the $q$ used for step $j\to j+1$}\\\hline 2&.14&.21&.142\\ 3&.12&.155&.0282\\ 4&.016&.026&.0028\\ 5&.001&.001&\text{---} \end{array} \tag{49}\] The exponents \((M_j-k_j)/M_j\) are \(98/240\), \(1/2\), \(1/2\). The successive upper outputs for \(E_{j+1}\) are below \(.117754,.015328,.000170\); those for \(\eta_{j+1}\) are below \(.147078,.025188,.000432\). In computing the latter outputs one uses the displayed rounded \(E_{j+1}\), so each line closes with the next.

For clarity, all the scalar comparisons here and in the periodic initialization have an elementary rational verification. For \(0\le t\le10\), put \(T_{80}(t)=\sum_{r=0}^{80}t^r/r!\). Then \[ T_{80}(t)\le e^t\le T_{80}(t)+2t^{81}/81!. \tag{50}\] Indeed the ratio of consecutive omitted terms is at most \(10/82<1/2\). For negative exponents use reciprocal intervals. Rational positive root comparisons can be checked by raising both sides to their integer powers; for example the first \(q\) bound in (49) is \(d(u_3)^{49}<.142^{120}\). Every denominator is positive. These rules, exact rational input values, and the displayed formulas suffice to verify the finite comparisons without floating-point assumptions. The supplementary scalar certificate records all of them.

Indefinite continuation

We now enlarge the three defect bounds at \(j=5\) to \(.01\) so that a single quadratic recurrence controls all later levels.

Reduction to fifteen thermal traces

For \(2\le n\le6\) and \(g\in\{1,-1,0\}\), define \[\begin{align*} H_{n,g} &=\sum_{j=0}^{n-2}\mathbf S_j\mathbin\cdot\mathbf S_{j+1} -\frac35S_0^z+\frac35R_g, & R_g&= \begin{cases} gS_{n-1}^z,&g=1,-1,\\ S_{n-1}^x,&g=0, \end{cases} \tag{61}\\ Z_{n,g}&=\mathop{\mathrm{Tr}}\exp[-b_0(H_{n,g}+anI)]. \end{align*}\] Thus \(Z_{n,1}=Y_n(b_0)\) and \(Z_{n,-1}=Y_n^{\rm s}(b_0)\). The first bound in Proposition 15 concerns \(Z_{6,1}\). For the second, average the right-field direction over the rotation group.

Let \(\mathcal G\) be the rotation group generated by the diagonal half-turns and the cyclic permutation of the coordinate axes, and let \[\Pi=\frac1{|\mathcal G|}\sum_{\gamma\in\mathcal G}U_\gamma, \qquad I(n)=\langle v_{b_0},X_{b_0}^n\Pi v_{b_0}\rangle.\] The group has order \(12\), and the orbit of \(z\) is the six signed coordinate axes, each with the same multiplicity. The boundary transfer identity therefore writes \(I(n)\) as the average of the six corresponding right-field partition functions. Global rotations about \(z\) identify the four perpendicular orientations. Hence \[ I(n)=\frac{Z_{n,1}+Z_{n,-1}+4Z_{n,0}}6. \tag{62}\] The averaging operator \(\Pi\) is an orthogonal projection commuting with \(X_{b_0}\). By Proposition 6, \(\mathcal G\) fixes the leading line pointwise. Therefore \[\begin{align*} c_0P_*(\lambda_0)^2 &\le\|P_*(X_{b_0})\Pi v_{b_0}\|^2\\ &=\langle v_{b_0},P_*(X_{b_0})^2\Pi v_{b_0}\rangle\\ &=I(6)-.28I(5)-.3404I(4)+.0504I(3)+.0324I(2). \tag{63}\end{align*}\] In particular, no positivity of the individual odd powers of \(X_{b_0}\) is being assumed.

It therefore suffices to enclose \(Z_{n,g}\) for \(2\le n\le6\) and \(g\in\{1,-1,0\}\). The remaining subsections provide these enclosures and evaluate (63).

An integer matrix for the thermal approximation

Set \[ C_n=3\left\lfloor\frac{n-1}{2}\right\rfloor +2\bigl((n-1)\bmod2\bigr)+\frac65, \qquad A=I-\frac{H_{n,g}+C_nI}{8}. \tag{64}\]

The operator \(A\) is a self-adjoint contraction. Here is a direct verification with the spin normalization in the theorem. A single bond has eigenvalues \(-2,-1,1\). For two consecutive bonds, couple the two outside spins to spin \(t\in\{0,1,2\}\) and then couple this with the middle spin to total spin \(j\). The operator \[\mathbf S_1\mathbin\cdot\mathbf S_2+ \mathbf S_2\mathbin\cdot\mathbf S_3 =\mathbf S_2\mathbin\cdot(\mathbf S_1+\mathbf S_3)\] has eigenvalues \([j(j+1)-t(t+1)-2]/2\), with \(|t-1|\le j\le t+1\). Its minimum is \(-3\). Pairing consecutive bonds, with one single bond left over if necessary, and using the total boundary-field norm \(6/5\) gives \[0\le H_{n,g}+C_nI \le\left(n-1+\frac65+C_n\right)I\le16I.\] Adding the lower bounds for the paired bonds is valid even where the three-site supports overlap. In order \(n=2,\ldots,6\), the displayed upper bounds are \(5.4,7.4,10.4,12.4,15.4\). Thus \(-I\le A\le I\); positivity of \(A\) itself is not required.

Index the spin-weight basis by words \(w=(w_0,\ldots,w_{n-1})\) in \(\{-1,0,1\}^n\), and let \[p_w=\left(\sum_{j=0}^{n-1}w_j\right)\bmod2\in\{0,1\}, \qquad J_{ww}=\sqrt{1+p_w},\qquad D=80JAJ^{-1}.\] The matrix \(D\) is integral. Indeed, its diagonal is \[ \eta(w)=80-10C_n-10\sum_{j=0}^{n-2}w_jw_{j+1} +6w_0-6g w_{n-1}. \tag{65}\] Each allowed bond hop \(w\leftrightarrow w+\sigma(\mathbf e_j-\mathbf e_{j+1})\), \(\sigma=\pm1\), has entry \(-10\): the corresponding entry of the spin-one bond is \(1\), and such a hop preserves \(p_w\). When \(g=0\), an allowed end hop from column \(r=w+\sigma\mathbf e_{n-1}\) to row \(w\) has entry \(-3(1+p_w)\). To see this coefficient, the end-field entry before similarity is \(3\sqrt2/10\), and the two parities are opposite, so \[-10\frac{3\sqrt2}{10} \sqrt{\frac{1+p_w}{1+p_r}}=-3(1+p_w).\] These are all the entries of \(D\).

The exact recurrence

The recurrence approximates the half-temperature exponential \[E=\exp[-b_0(H_{n,g}+C_nI)/2]=e^{-49}\exp(49A).\] For a finite matrix \(M\), write \(\|M\|_{\rm F}^2=\mathop{\mathrm{Tr}}(M^*M)\) for its squared Frobenius norm. Since \(E\) is self-adjoint, \(\|E\|_{\rm F}^2\) is the thermal trace \(\mathop{\mathrm{Tr}}\exp[-b_0(H_{n,g}+C_nI)]\). We approximate \(E\) by a normalized Poisson polynomial in the contraction \(A\); the similarity \(D=80JAJ^{-1}\) permits an integer Horner recurrence.

Put \[ K=192,\qquad Q=2^{50},\qquad s_i=\frac{49^iK!}{i!},\qquad S_K=\sum_{i=0}^Ks_i, \qquad t_i=\left\lfloor\frac{Qs_i}{S_K}\right\rfloor. \tag{66}\] Starting with the zero matrix \(\mathcal X_{K+1}\), compute \[ \mathcal X_i= \left\lfloor\frac{D\mathcal X_{i+1}}{80}\right\rfloor+t_iI, \qquad i=K,K-1,\ldots,0. \tag{67}\] Matrix floors are entrywise, and every floor, including one with a negative argument, is toward \(-\infty\). Define \[ \widehat E=J^{-1}\frac{\mathcal X_0}{Q}J, \qquad W_{n,g}=\|\widehat E\|_{\rm F}^2. \tag{68}\]

For completeness, the following scalar version specifies the recurrence without matrix construction. Number words, if desired, by \(\sum_j(w_j+1)3^j\), and interpret a lookup outside \(\{-1,0,1\}^n\) as zero. Initialize \(x(w,r)=0\) for all pairs of words. At stage \(i=K,\ldots,0\), use the current array to form \[\begin{align*} z(w,r)={}&\eta(w)x(w,r) -10\sum_{j=0}^{n-2}\sum_{\sigma=\pm1} x\bigl(w+\sigma(\mathbf e_j-\mathbf e_{j+1}),r\bigr) \\ &\hspace{5mm} -\mathbf1_{\{g=0\}}\,3(1+p_w) \sum_{\sigma=\pm1}x(w+\sigma\mathbf e_{n-1},r), \tag{69}\\ x_{\rm new}(w,r)={}&t_i\mathbf1_{\{w=r\}} +\left\lfloor\frac{z(w,r)}{80}\right\rfloor. \end{align*}\] Replace all entries simultaneously. At the end, the required table entry is the integer quotient \[ k_{n,g}:=\left\lfloor10^{16}W_{n,g}\right\rfloor =\left\lfloor \frac{10^{16}\displaystyle\sum_{w,r} x(w,r)^2(2-p_w)(1+p_r)}{2Q^2} \right\rfloor. \tag{70}\] The weight follows from \((1+p_r)/(1+p_w)=(2-p_w)(1+p_r)/2\). All operations in (66), (69), and (70) are integer operations; they may be performed with arbitrary-precision integers.

The resulting fifteen entries are \[ \begin{array}{c|rrr} n\backslash g&1&-1&0\\\hline 2&10185296075298&4160017987&583578389575\\ 3&259254948723&571111660440989&47796720049308\\ 4&318278534893&661347985&43971049724\\ 5&260908126900&29917222640283&5659378873682\\ 6&16467340956&328219854&4086161507 \end{array} \tag{71}\] Equations (66), (69), and (70) provide a finite exact-arithmetic verification of every entry.

Error of the matrix approximation

Squaring the Frobenius norm converts this estimate into a thermal trace. In fact, \(E\) is self-adjoint and \(\|E\|_{\rm F}^2=\mathop{\mathrm{Tr}}\exp[-b_0(H_{n,g}+C_nI)]\). Consequently \[ \left|Z_{n,g}-e^{b_0(C_n-an)}W_{n,g}\right| <e^{b_0(C_n-an)} \left(4\cdot10^{-10}\sqrt{W_{n,g}}+4\cdot10^{-20}\right). \tag{74}\] This follows from \(|\|E\|_{\rm F}^2-\|\widehat E\|_{\rm F}^2| \le2\|E-\widehat E\|_{\rm F}\|\widehat E\|_{\rm F} +\|E-\widehat E\|_{\rm F}^2\).

Rational intervals from the printed table

The displayed integer floors suffice to verify the needed estimates; no unprinted digits of \(W_{n,g}\) are used. For an entry \(k=k_{n,g}\), put \[ L=\frac{k}{10^{16}},\qquad U=\frac{k+1}{10^{16}},\qquad r=\frac{\lfloor\sqrt{k+1}\rfloor+1}{10^8},\qquad \varepsilon=4\cdot10^{-10}r+4\cdot10^{-20}. \tag{75}\] Then \(L\le W_{n,g}<U\) and \(\sqrt{W_{n,g}}<r\). In every entry \(L-\varepsilon>0\), so (74) gives \[ e^{b_0(C_n-an)}(L-\varepsilon) <Z_{n,g}<e^{b_0(C_n-an)}(U+\varepsilon). \tag{76}\] The square root in (75) is just an integer square root. To enclose the remaining exponential by rational numbers, set \[T_{80}(y)=\sum_{j=0}^{80}\frac{y^j}{j!}.\] For \(0\le y\le10\), comparison of consecutive tail terms gives \[ T_{80}(y)\le e^y \le T_{80}(y)+\frac{2y^{81}}{81!}. \tag{77}\] For a negative argument, take reciprocals and reverse the endpoints. All arguments \(b_0(C_n-an)\) here lie between \(-.0544635\) and \(9.691073\). Thus (75)–(77) require only rational arithmetic and integer square roots. In particular they give \[ .0266319440<Z_{6,1}<.0266319607<.027. \tag{78}\]

Using (76) and (77) in (62) gives the following rational centers, each with absolute error less than \(10^{-7}\): \[ \begin{array}{c|ccccc} n&2&3&4&5&6\\\hline \text{center of }I(n) &.02703074&.01203562&.01193511&.00931560&.00893272. \end{array} \tag{79}\]

Substitution of these centers in the right-hand side of (63) gives \(.003744031780\). The sum of the absolute polynomial coefficients is \(1.7032\), so the total possible error is less than \(1.7032\cdot10^{-7}\). The right-hand side of (63) is consequently less than \[.003744031780+.00000017032=.0037442021<.003745.\] Together with (78), this proves Proposition 15.

The complete tensor and endpoint data

Let \(\mathcal V\) have the \(26\) labels \[ (i,d),\qquad 1\le i\le8,\qquad d=-\ell_i,-\ell_i+2,\ldots,\ell_i, \qquad(\ell_i)_{i=1}^8=(1,1,1,1,3,3,3,5). \tag{82}\] Order them first by \(i\) and then by increasing \(d\). Three real \(26\)-by-\(26\) matrices \(A_s\), \(s\in\{-1,0,1\}\), are specified as follows. At row \((i,d)\) and column \((j,e)\) put \(\ell=\ell_i\), \(k=\ell_j\), and \(r=(\ell+d)/2\). The entry is zero unless \[ e=d+2s,\qquad |k-\ell|\le2,\qquad k=\ell\ \Longrightarrow\ i=j. \tag{83}\] For an allowed entry, multiply the coefficient in the second column of Table [apptr:entry-table] by the factor in its \(s\) column. The parameters \(D_i\) and \(P_{ij}\) appear in Tables [apptr:endpoint-table] and 3. The latter parameter is defined only when \(\ell_j=\ell_i+2\). All labels in these rules must belong to (82); an out-of-range column is absent.

Define columns \(\mathbf b,\mathbf c\in\mathbb R^{26}\) by \[ \mathbf b_{(i,d)}=\begin{cases}\beta_i,&d=-1,\\0,&d\ne-1,\end{cases} \qquad \mathbf c_{(i,d)}=\begin{cases}\kappa_i\beta_i,&d=-1,\\0,&d\ne-1.\end{cases} \tag{84}\] All their parameters are included in Table [apptr:endpoint-table].

For each \(m\ge2\), the physical vector is \[ \psi_m=\sum_{s_1,\ldots,s_m=-1}^{1} \bigl(\mathbf b^{\mathsf t}A_{s_1}\cdots A_{s_m}\mathbf c\bigr) e_{s_1}\otimes\cdots\otimes e_{s_m}. \tag{85}\] This is the open-boundary matrix-product form of (Pérez-García et al. 2007, sec. 3.1, Equation (11)), with endpoint rows and columns absorbed into the first and last matrices. The vectors \(e_{-1},e_0,e_1\) are the orthonormal spin basis fixed in the statement of the problem. In particular, the coefficients in (85) are ordinary real amplitudes: no virtual inner product or physical basis weights are inserted. Normalization of the matrices \(A_s\) is unnecessary. The support rule forces a nonzero amplitude to have \(\sum_j s_j=0\), because both endpoints have virtual weight \(-1\). This restriction on the trial vector does not restrict the variational upper bound on the smallest eigenvalue of the full Hamiltonian.

The following tables complete the integer specification of these matrices and endpoints.

Norm, fields, and the bond insertion

For \(s,t\in\{-1,0,1\}\), put \(B_{st}=A_sA_t\) and define \[\begin{align*} T_*&=\sum_{s=-1}^{1}A_s\otimes A_s, &G_*&=\sum_{s=-1}^{1}sA_s\otimes A_s, \tag{86}\\ O_*&=\sum_{s,t=-1}^{1}st\,B_{st}\otimes B_{st} \\[-2pt] &\quad+ \sum_{\substack{s=-1,0\\t=0,1}} \bigl(B_{st}\otimes B_{s+1,t-1} +B_{s+1,t-1}\otimes B_{st}\bigr). \tag{87}\end{align*}\] We use the usual Kronecker convention \((C\otimes D)_{(v,w),(v',w')}=C_{v,v'}D_{w,w'}\). Write \[ L_0=(\mathbf b\otimes\mathbf b)^{\mathsf t}, \qquad u=\mathbf c\otimes\mathbf c, \qquad L_i=L_0T_*^i. \tag{88}\]

The invariant 162-dimensional contraction

All contractions in Lemma 18 can be carried out in a single \(162\)-dimensional space. On a paired virtual label \(((i,d),(j,e))\), consider the difference \(d-e\). Each nonzero entry of \(A_s\) changes virtual weight by \(2s\) from row to column. Consequently every summand \(A_s\otimes A_s\) of \(T_*\) or \(G_*\) preserves this difference. A nonzero entry of \(B_{st}\) changes weight by \(2(s+t)\). The two factors in every tensor summand of \(O_*\) have the same total physical weight, including the pair \((s,t)\) and \((s+1,t-1)\) in the off-diagonal terms. Each such summand also preserves \(d-e\).

Let \(\mathcal K_0\) be the coordinate subspace with paired labels \[ \mathcal Q=\{((i,d),(j,d)):(i,d),(j,d)\in\mathcal V\}. \tag{91}\] The multiplicities of the virtual weights \(d=-5,-3,-1,1,3,5\) are \(1,4,8,8,4,1\), respectively; hence \[ \dim\mathcal K_0=1^2+4^2+8^2+8^2+4^2+1^2=162. \tag{92}\] The preceding argument shows that \(T_*,G_*,O_*\) are block diagonal by weight difference. Both \(L_0\) and \(u\) are supported in \(\mathcal K_0\), because the two endpoint vectors are supported on \(d=-1\). We may therefore restrict these three matrices to (91) before multiplying them, taking powers, or forming the expressions of Lemma 18. In particular, squaring the restricted \(T_*\) gives the restriction of \(T_*^2\); there is no contribution through a discarded state. From now on the same symbols denote these restricted integer matrices and endpoint tensors, ordered by the inherited lexicographic order.

An exact row recurrence

The energy sum in (90) need not be evaluated separately at every bond. Define integer rows by \[\begin{align*} L_i&=L_{i-1}T_*\quad(i\ge1),\\ N_1&=-300000L_0G_*,\\ N_i&=N_{i-1}T_*+ L_{i-2}\bigl(500000O_*+700741T_*^2\bigr) \quad(i\ge2). \tag{93}\end{align*}\] Then \[ V_m=L_mu,\qquad H_m^*=\bigl(N_m+300000L_{m-1}G_*+700741L_m\bigr)u. \tag{94}\] To verify the latter identity, induction on \(m\) in (93) gives \[ N_m=-300000L_0G_*T_*^{m-1} +\sum_{r=0}^{m-2} L_0T_*^r(500000O_*+700741T_*^2)T_*^{m-2-r}. \tag{95}\] Its bond contributions and left field are exactly those in (90). Each of its \(m-1\) shift contributions equals \(700741L_m\). The final term \(700741L_m\) in (94) supplies the remaining site shift, and the other final term supplies the right field with the required positive sign. This proves (94).

For specificity, the whole contraction can be implemented by keeping just three rows. Begin with \[x=L_0,\qquad y=L_0T_*,\qquad w=-300000L_0G_*.\] For each \(m=2,3,\ldots,240\), make the following updates, whose right-hand sides all use the old rows: \[\begin{align*} w_{\mathrm{new}}&=wT_*+500000xO_*+700741(xT_*)T_*,\\ x_{\mathrm{new}}&=y,\qquad y_{\mathrm{new}}=yT_*. \tag{96}\end{align*}\] After this iteration, \((x,y,w)=(L_{m-1},L_m,N_m)\). At a required length compute \[ V=yu,\qquad H=wu+300000x(G_*u)+700741V. \tag{97}\] Thus \((V,H)=(V_m,H_m^*)\). The initialization realizes the same invariant at \(m=1\), which proves the indexing of the algorithm. Ordinary exact integer matrix multiplication suffices throughout; sparsity can reduce the work but changes no operation or comparison.

The integer comparisons

Table 4 gives the results of (96)–(97). The integers \(p_m\), \(q_m\), and \(k_m\) in the table have the following exact meanings: \[\begin{align*} 10^{p_m-1}&\le V_m<10^{p_m},\\ q_m10^{p_m-6}&\le V_m<(q_m+1)10^{p_m-6},\\ k_mV_m&\le20H_m^*<(k_m+1)V_m. \tag{98}\end{align*}\] The last comparison is equivalent to \(k_m=\lfloor10^7H_m^*/(500000V_m)\rfloor\), since \(10^7/500000=20\). The first comparison in particular certifies that the vector \(\psi_m\) is nonzero.

All four strict comparisons \[ 202100V_m-H_m^*>0 \qquad(m=34,78,142,240) \tag{99}\] follow as well. They are recorded directly in the full integer output, and they also follow from the last line of (98), since every displayed \(k_m+1\) is less than \(4042000=20\cdot202100\).

The accompanying computational files make the integer comparison fully reproducible. The program constructs the matrices from the data in this appendix and executes the row recurrence with arbitrary-precision integers. The file contains every entry of \(A_s\), the endpoint vectors, the restricted paired labels, and the matrices \(T_*,G_*,O_*\). The file contains the full integers \(V_m\), \(H_m^*\), and \(202100V_m-H_m^*\) for each row, rather than only their shortened display in Table 4. The equations and tables of this appendix specify the calculation independently of these files.