Historical context

The half-odd-integer side has a rigorous antecedent in the low-energy excitation bound of Lieb, Schultz and Mattis for the cyclic spin-one-half chain (Lieb et al. 1961, Appendix B, Theorem 2). Affleck and Lieb extended this obstruction to the half-odd-integer antiferromagnetic chains they studied (Affleck and Lieb 1986). The possible alternative of degenerate ground states is part of that obstruction. Integer spin permits a unique gapped state, but this distinction alone does not prove a gap for a particular interaction.

The Affleck–Kennedy–Lieb–Tasaki construction established a rigorous gapped spin-one chain with a biquadratic interaction (Affleck et al. 1987, 1988). Its bond interaction is \(h+\tfrac13h^2\), up to a scalar and normalization, where \(h=\mathbf S_j\cdot\mathbf S_{j+1}\). The Hamiltonian in (1) has bond interaction \(h\) itself. Yarotsky proved that a unique ground state and a uniform periodic gap persist under sufficiently small translation-invariant finite-range perturbations of the AKLT interaction (Yarotsky 2006, arXiv v1, Theorems 1 and 3). This controls a neighborhood of that interaction; it does not establish a gapped path to the pure bilinear point.

For the pure spin-one chain, White and Huse’s density-matrix renormalization-group study estimated a bulk gap of \(0.41050(2)\) and exhibited effective spin-one-half degrees of freedom at open ends (White and Huse 1993). Neutron scattering provided experimental evidence in the nearly one-dimensional spin-one antiferromagnet \(\mathrm{CsNiCl}_3\) (Buyers et al. 1986). These results explain both the strong physical evidence for the prediction and the significance of boundary conditions. The uniform lower bounds below instead follow from the stated finite certificates and an analytic argument valid at every larger even length.

Batista and Ortiz proposed a spin–fermion proof of the Haldane gap (Batista and Ortiz 2001). Their gap argument following Equation (8) of the preprint version uses a first-order perturbative identification with an XYZ chain; that step does not supply a uniform remainder estimate or a continuation to the original pairing coefficient. More recently, Tasaki established a nontrivial infinite-chain topological index for the spin-one Heisenberg model under a gap assumption for odd open chains with endpoint fields (Tasaki 2025, arXiv v4, Assumption 2 and Theorem 3). The boundary hypotheses of that theorem require a separate argument from the even-periodic estimates proved here.

Finite-volume methods for uniform spectral gaps have a substantial history, including the criteria of Knabe and Nachtergaele for frustration-free chains, whose ground states minimize every local interaction (Knabe 1988; Nachtergaele 1996). Gosset and Mozgunov obtained sharper finite-size thresholds within that frustration-free setting (Gosset and Mozgunov 2016). Recently, Banerjee, Guo and Chowdhury developed semidefinite bounds for string correlators using an enclosing ground-energy window, including finite periodic spin-one Heisenberg chains (Banerjee et al. 2026, Equations (2), (13) and Figure 4). Those observable bounds concern a different quantity from the uniform spectral gap studied here. Our finite inputs are thermal traces of small twisted chains and variational energy estimates from periodic matrix-product vectors, in the tradition of finitely correlated states (Fannes et al. 1992). The latter use the standard trace-of-matrix-products representation (Pérez-García et al. 2007, sec. 2.1); we give the particular integer matrices and all contractions explicitly.

The finite thermal inputs are rigorous enclosures of matrix-exponential traces. Validated matrix exponentiation has a long history, including self-validating Padé methods (Bochev and Markov 1989) and interval polynomial evaluation (Goldsztejn and Neumaier 2014). Our particular implementation uses a conditioned Poisson polynomial in a contraction. The factor-two conditioning estimate follows the coefficient argument of Fox and Glynn (Fox and Glynn 1988, Proposition 1). Explicit bounds for the integer evaluation and the final trace enclosure are supplied in the appendices.

How the proof works

Quantum transfer-matrix methods relate partition functions to spatial moments (Suzuki 1985). Our construction uses an ordered bond expansion of the type developed by Aizenman and Nachtergaele (Aizenman and Nachtergaele 1994, arXiv v1, Section 2.2), followed by a direct operator construction on bond histories.

A shifted partition function is a sum of positive physical Boltzmann weights. We construct a self-adjoint spatial transfer operator whose \(L\)th moment is the same partition function. For even \(L\), the spatial moment is also a sum of nonnegative weights. This gives two probability distributions associated with one partition function. The purity of a distribution \((p_i)\) is \(\sum_i p_i^2\); purity close to one means that a single weight carries almost all of its mass.

Squaring and renormalizing a distribution reduces its purity defect quadratically once the defect is small. Squaring physical weights doubles inverse temperature, while squaring spatial weights doubles length. Exact cancellation identities couple these two improvements. Starting from two finite purity bounds, both defects then decay doubly exponentially as length and inverse temperature double together. Spatial moment inequalities cover every intermediate even length. Finally, purity of the full physical Gibbs distribution bounds each excited-to-ground probability ratio and hence the physical energy gap. Proposition 8 states this coupled argument as a general gap criterion. Its assumptions isolate a reusable criterion for other Hamiltonian families with the same two partition-function representations; positivity of the spatial transfer is not required.

There are two finite initializations. One starts the iteration at length \(60\) and inverse temperature \(105/4\). The other reaches length \(2304\) and inverse temperature \(784\) with a smaller purity defect, giving the stronger eventual bound. They share the transfer construction and the bootstrap, while using different thermal moments and two lengths of one matrix-product trial family. The finite computations involve integer arithmetic, rational interval bounds and explicit truncation errors; no floating-point eigenvalue estimate is a premise.

Section 2 fixes the model and proves its elementary energy bounds. Section 3 constructs the spatial transfer and its rotation sectors. Section 4 proves the common purity theorem, and Section 5 establishes the two finite initializations from explicitly stated thermal and trial inputs. Section 6 combines these results to prove Theorem 1. Appendices 7–9 prove the essential thermal and trial inputs, with exact data and reproducible finite procedures.

Spin coordinates

We use the self-adjoint spin-one matrices. This convention does not restrict the spectrum in the formulation where self-adjointness is not assumed.

In the weight basis \(|s\rangle\), \(s\in\{-1,0,1\}\), the diagonal generator is \(S^z|s\rangle=s|s\rangle\) and the nonzero ladder entries are \(\sqrt2\). We will also use the Cartesian basis, indexed by the three axes, in which \[(S^\alpha)_{ij}=-i\varepsilon_{\alpha i j},\qquad G_\alpha=iS^\alpha.\] These matrices satisfy the same relations. Each \(G_\alpha\) is real and has operator norm one. For the two-site interaction \(h=\sum_\alpha S^\alpha\otimes S^\alpha\), contraction of the epsilon tensors gives \[ -h=\sum_\alpha G_\alpha\otimes G_\alpha, \qquad h=F-|\Omega\rangle\langle\Omega|, \qquad \Omega=\sum_{i=1}^3|ii\rangle, \tag{6}\] where \(F(u\otimes v)=v\otimes u\). In particular, \(h\) has eigenvalues \(-2,-1,1\), and \(h\le I\).

Twists and partition functions

Number the sites \(0,\ldots,n-1\), with \(n\ge4\). A proper Cartesian rotation \(g\in SO(3)\) is unitary on \(\mathbb C^3\). Define \[ H_n(g)=\sum_{j=0}^{n-2}h_{j,j+1}+h_{n-1,0}(g), \qquad h_{n-1,0}(g)=\sum_\alpha S_{n-1}^\alpha(gS^\alpha g^{-1})_0. \tag{7}\] Thus only the last bond is conjugated, on its site-\(0\) factor. The Hamiltonian is self-adjoint, and \(H_n(I)=H_n\). Fix \[ a=\frac{700741}{500000},\qquad Z_n(b,g)=\mathop{\mathrm{Tr}}\exp[-b(H_n(g)+anI)],\qquad Z_n(b)=Z_n(b,I) \quad (b>0). \tag{8}\] All these partition functions are strictly positive. The scalar shift will cancel from the purity ratios and from the final spectral gap. With this normalization, the trial bounds in Lemma 17 give \(E_0(n)+an<0\) at \(n=72,120\), and hence \(Z_n(b)>1\) for every \(b>0\). These lower bounds are paired with finite thermal upper bounds in Section 5.

We need the diagonal rotations \[P_x=\operatorname{diag}(1,-1,-1),\quad P_y=\operatorname{diag}(-1,1,-1),\quad P_z=\operatorname{diag}(-1,-1,1),\qquad D_2=\{I,P_x,P_y,P_z\},\] and the cyclic rotation \(C:e_x\mapsto e_y\mapsto e_z\mapsto e_x\). Write \(P=P_x\). The three nonidentity elements of \(D_2\) have equal partition functions, by a global cyclic rotation of all sites. The rotation \(C\) has angle \(2\pi/3\) about the diagonal axis. After a global rotation and a change to the weight basis, \(P\) and \(C\) are respectively represented by \(\exp(-i\theta S^z)\) with \(\theta=\pi\) and \(\theta=2\pi/3\).

For later comparison with finite matrices, a word \(x=(x_0,\ldots,x_{n-1})\) in the weight basis has diagonal entry \(\sum_j x_{j-1}x_j\), with indices modulo \(n\). Each permitted step changing two neighboring letters by opposite units has coefficient one. For a seam step from column \(y\) to row \(x\), this coefficient is \[ \exp\bigl(i\theta(y_0-x_0)\bigr). \tag{9}\] Indeed \(gS^\pm g^{-1}=e^{\mp i\theta}S^\pm\). These formulas identify the finite matrices used below with the physical Hamiltonians in (7).

A uniform energy bound

Rotation sectors

The small twisted partition functions separate the spatial eigenvalues into a few lists. Their multiplicities will help isolate a dominant weight; no restriction on the physical spectrum is imposed.

The invariant \(I\) space need not be one-dimensional. The point of Lemma 5 is that every weight outside it occurs at least twice in the full list. Later moment bounds will identify a unique largest weight without discarding any sector.

The coupled update

The two purities constrain one another because their definitions use the same partition functions. Direct cancellation gives \[\begin{align*} S(n,2b)&=S(n,b)^2\frac{T(2n,b)}{T(n,b)^2}, \tag{26}\\ T(4n,b)&=T(2n,b)^2\frac{S(2n,2b)}{S(2n,b)^2}. \tag{27}\end{align*}\] Both denominators are at most one. The following form allows the initial bounds to have different sizes and to be rounded upward.

In particular, one may round the first output upward before using it in the second update, then round the second output upward. The pair based at \((n,b)\) becomes a pair based at \((2n,2b)\): its physical member is at length \(4n\). No decrease of the individual bounds is needed for this step.

Quadratic decay and intermediate lengths

Once both defects are small, one common bound suffices. Put \[ G(u)=\frac12\left((1-u)^{-1}+(1-u)^{-2}+(1-u)^{-3}\right)^2. \tag{29}\] This function is increasing on \([0,1)\), and \[ f\bigl(1-(1-u)^3\bigr)=G(u)u^2. \tag{30}\]

Figure 1 summarizes the two operations in the proof: doubling advances the pair of purity estimates, and interpolation fills the interval of physical lengths controlled at each temperature.

The two parameter choices

We will establish the initial inequalities (32) for the following two rows: \[ \begin{array}{c|c|c|c|c} n_0&b_0&u_0&C&r=Cu_0\\ \hline 60&105/4&1/8&79/10&79/80\\ 2304&784&1/100&5&1/20 \end{array} \tag{39}\] The scalar hypotheses of Proposition 8 can already be checked. For the first row, \[G(1/8)=\frac{913952}{117649}<\frac{79}{10},\qquad \frac{3}{C(1-3u_0)}=\frac{48}{79}<1.\] For the second, \(1-(1-u)^3\le3u\) gives, when \(0<u\le1/100\), \[G(u)\le\frac{9}{2(1-3u)^2} \le\frac{45000}{9409}<5,\qquad \frac{3}{C(1-3u_0)}=\frac{60}{97}<1.\] Thus the two rows give respectively the bounds \((4/105)\log(80/79)\) and \((\log20)/784\) in (35), once their initial purities are proved. The second row will follow from six applications of Lemma 7 to bounds based at \((36,49/4)\). The remaining task is therefore finite: establish the two sets of initial inequalities without losing the margins needed in these updates.

Thermal data and elementary moment bounds

We use the sectors from Lemma 5: \(J_n=I_n+2O_n\) is the \(D_2\)-invariant moment, \(I_n\) is the moment fixed also by the cyclic rotation, and \(N_n\) is the moment in one nontrivial \(D_2\) sector. Each \(O_n\) occurs twice and each \(N_n\) occurs three times in the full trace. In particular, \[ \begin{aligned} J_n&=\frac{Z_n+3Z_n(P)}4,& N_n&=\frac{Z_n-Z_n(P)}4,\\ I_n&=\frac{Z_n+3Z_n(P)+8Z_n(C)}{12},& O_n&=\frac{Z_n+3Z_n(P)-4Z_n(C)}{12}. \end{aligned} \tag{40}\] The inverse temperature is common to every term of each identity.

Appendices 7 and 8 prove the respective tables by rational approximations to the physical matrix exponentials, with explicit error bounds. Lemma 17 supplies the other finite input: \[ E_0(72)<-72a,\qquad E_0(120)<-120a. \tag{41}\] Consequently \(Z_{72}(b)>1\) and \(Z_{120}(b)>1\) for every \(b>0\): in each case one Boltzmann factor already exceeds one.

Here are two elementary ways to extract spectral information from the tables.

All moments used below converge absolutely by Proposition 4. The coefficient sums in (40) are at most one in absolute value, so the thermal error bound also bounds each sector-moment error. This observation includes the odd moments used in the second initialization; their signs are retained in the polynomial filter.

The initialization at length sixty

The initialization giving the stronger gap

A periodic interface for boundary estimates

The boundary-field companion (OpenAI 2026) uses the intermediate spatial information in the preceding proof, rather than only the final periodic gap. We state that information separately so that its hypotheses and finite inputs can be invoked together.

An infinite-volume consequence

The finite-volume estimate also controls local excitations in a thermodynamic limit. We use the standard limiting argument recalled in (Tasaki 2025, arXiv v4, End Matter, Definition 11 and Theorem 12). A local operator on the spin chain \(\mathbb Z\) acts nontrivially on only finitely many sites. A state \(\omega\) is a linear functional on these operators with \(\omega(I)=1\) and \(\omega(A^*A)\ge0\) for every local operator \(A\). Define the local commutator \[[H,A]=\sum_{j\in\mathbb Z} [\mathbf S_j\cdot\mathbf S_{j+1},A].\] Only finitely many summands are nonzero.

In particular, \(\omega\) is a ground state: no local perturbation lowers its energy. For \(\omega(A)=0\), inequality (48) is the locally unique gapped property of (Tasaki 2025, arXiv v4, Definition 11), with gap at least \(\log(20)/784\). The argument does not identify the limits of different subsequences or establish uniqueness among all infinite-volume ground states.

The finite matrices and their columns

Fix \(b=21/2\) and one of these lengths, and write \(H=H_n(g)\). After the global rotations in Section 2, the three twists have angles \(\theta=0,\pi,2\pi/3\) about the weight-basis axis. Set \[ W=I-\frac{H_n(g)+(3n/2)I}{16},\qquad B=16W. \tag{49}\] Lemma 3 gives \(\left\lVert W\right\rVert\le1\) and \(\left\lVert B\right\rVert\le16\). In a fixed total-weight sector, the rows and columns are words \(x\in\{-1,0,1\}^n\) with fixed \(\sum_jx_j\). The diagonal of \(B\) is \[B_{x,x}=16-\frac{3n}{2}-\sum_{j=0}^{n-1}x_{j-1}x_j.\] For each \(j\), put \(k=j+1\pmod n\). If \(y_j=x_j-\delta\), \(y_k=x_k+\delta\), \(\delta\in\{-1,1\}\), and all letters remain in \(\{-1,0,1\}\), the corresponding entry is \(-1\), except at \(j=n-1,k=0\), where it is \(-e^{i\theta\delta}\). This is exactly the physical seam phase in (9). For the cyclic twist its action on a coordinate \(u+v\omega\) uses \[ -\omega(u+v\omega)=v+(v-u)\omega, \qquad -\omega^{-1}(u+v\omega)=(u-v)+u\omega. \tag{50}\] Thus matrix-vector multiplication requires only integer additions and multiplications in both coordinates.

Only some columns need to be computed. To justify the reduction, let \(s(x)=(x_{n-1},x_0,\ldots,x_{n-2})\). The seam formula gives \[H_{s(x),s(y)} =e^{i\theta(y_{n-1}-x_{n-1})}H_{x,y}.\] Consequently the monomial unitary \(|x\rangle\mapsto e^{-i\theta x_{n-1}}|s(x)\rangle\) commutes with \(H\). Reversal of the word takes \(H\) to its complex conjugate, as does negation of every letter. Composing either permutation with complex conjugation gives an antiunitary symmetry. Each of these symmetries preserves the norm of the corresponding column of any real exponential of \(H\).

In every sector of nonnegative total weight, select the lexicographically least word in each orbit under rotations and reversal. Give it the orbit size as its multiplicity, doubled for strictly positive total weight to include the opposite sector. The multiplicities then sum to \(3^n\) and reproduce the full Frobenius norm of the exact exponential. No symmetry of the rounded columns is required: we compare each computed representative with its own exact column and repeat that comparison with the same multiplicity.

A rational polynomial and its error

Write \[E=e^{84(W-I)}=e^{-(b/2)(H_n(g)+(3n/2)I)}, \qquad x_n=(3/2-a)bn.\] The desired trace is \[ Z_n(b,g)=e^{x_n}\left\lVert E\right\rVert_F^2, \tag{51}\] where \(\left\lVert \cdot\right\rVert_F\) is the Frobenius norm. For \(0\le j\le J=256\), define the rational probabilities \[ p_j=\frac{84^j/j!}{\sum_{l=0}^{256}84^l/l!}, \qquad P(W)=\sum_{j=0}^{256}p_jW^j. \tag{52}\] These are Poisson probabilities of mean \(84\), conditioned on values at most \(256\). For a Poisson variable \(V\) with that mean, \[\Pr(V>256)\le\frac{\mathbb E(2^V)}{2^{257}} =\frac{e^{84}}{2^{257}} <\frac{27^{28}}{2^{257}}<2^{-117}.\] The conditioning estimate is the coefficient argument of Fox–Glynn (Fox and Glynn 1988, Proposition 1), applied here to powers of a contraction. Since \(\left\lVert W^j\right\rVert\le1\), conditioning changes the operator average by at most twice this probability. Hence \[ \left\lVert P(W)-E\right\rVert<2^{-116}. \tag{53}\]

Put \(Q=2^{56}\) and \(q_j=\lfloor Qp_j\rfloor\). For a representative column \(r\), start at zero and apply, for \(j=256,\ldots,0\), \[ z\longleftarrow\operatorname{fl}(Bz/16)+q_j e_r. \tag{54}\] For real coordinates, \(\operatorname{fl}\) is the ordinary floor; for \(u+v\omega\) it floors \(u\) and \(v\) separately. The ideal suffix at step \(j\) is \(Q\sum_{l=j}^{256}p_lW^{l-j}e_r\), whose norm is at most \(Q\). In a sector of dimension \(d_0\), the Euclidean norm of the error from a coordinatewise floor is less than \(2\sqrt{d_0}\), and the coefficient floor adds at most one. Contraction of \(W\) therefore bounds the error at every stage by \[ 257(2\sqrt{3^n}+1)\le125159. \tag{55}\] The zero coefficients above \(q_{172}\) may be omitted: the computed vector is still zero there, and (55) continues to count all \(257\) possible rounding steps.

The norm of the repeated computed-column list, divided by \(Q\) and multiplied by \(e^{x_n/2}\), differs from \(\sqrt{Z_n(b,g)}\) by at most \[ e^{x_n/2}\sqrt{3^n} \left(\frac{257(2\sqrt{3^n}+1)}Q+2^{-116}\right) <4\cdot10^{-7}. \tag{56}\] Indeed \(\sqrt{3^n}\le243\), \(x_n/2<6\), and \(e^{x_n/2}<3^6\). The resulting rational upper bound \(3^6\,243(125159/2^{56}+2^{-116})\) is less than \(4\cdot10^{-7}\). This comparison includes both polynomial truncation and integer rounding.

Exact evaluation by packed integers

For efficiency, the calculation performs (54) on batches of at most \(192\) columns. A list of signed integer digits \(y_0,\ldots,y_{w-1}\) is stored as \[\operatorname{pack}(y)=\sum_{t=0}^{w-1}y_t2^{64t}.\] Integer linear arithmetic preserves this identity before the floor, even if intermediate carries occur. Arbitrary-precision integers are used throughout.

Here is why the floor can also be performed on the packed integer. For either coefficient of \(u+v\omega\), \[|u|,|v|\le\frac2{\sqrt3}|u+v\omega|\le2|u+v\omega|.\] By (55), every computed column has norm at most \(Q+125159\). Thus every coordinate of \(Bz\), in either integer component, has magnitude at most \[ 32(Q+125159)=2305843009217699040<2^{63}. \tag{57}\] Let \(s_w=\sum_{t=0}^{w-1}2^{64t}\) and define \[\mathrm{bias}=2^{63}s_w,\qquad \mathrm{mask}=(2^{60}-1)s_w,\qquad \mathrm{bias16}=2^{59}s_w.\] For \(Y=\operatorname{pack}(y)\) satisfying (57), the operation \[\texttt{(((Y+bias)>{}>4)\&mask)-bias16}\] equals \(\operatorname{pack}((\lfloor y_t/16\rfloor)_{t=0}^{w-1})\). To see this, adding the bias makes every base-\(2^{64}\) digit lie in \([0,2^{64})\). The global right shift moves four low bits from each higher lane into the four high bits of the preceding lane. The mask deletes precisely those crossing bits, retaining \(\lfloor(y_t+2^{63})/16\rfloor\) in lane \(t\). Subtracting \(2^{59}\) then gives \(\lfloor y_t/16\rfloor\), also for negative \(y_t\). This proves by induction that the packed recurrence is exactly (54).

The final coordinates satisfy the smaller bound \(2(Q+125159)<2^{63}\), so biasing and unpacking recovers every integer coordinate exactly. If a final coordinate is \(u+v\omega\), its squared modulus is \(u^2-uv+v^2\). Summing these integers with the representative multiplicities gives an integer \(t_{n,g}\), the squared norm of the full repeated-column list in \(Q\)-units.

The exact totals and trace enclosures

Integer evaluation of the recurrence gives the following totals. The accompanying computation implements the matrix entries, representatives and packed operations specified above.

To enclose the scalar factor, put \[L_n=\sum_{j=0}^{100}\frac{x_n^j}{j!},\qquad U_n=L_n+2\frac{x_n^{101}}{101!}.\] Since \(0<x_n<12\) and successive terms after the first omitted term have ratio at most \(x_n/102<1/2\), we have \(L_n<e^{x_n}<U_n\). For each center \(c_{n,g}\) in the \(b=21/2\) table of Lemma 9, exact rational comparison gives \[ c_{n,g}-2\cdot10^{-6} <\frac{L_n t_{n,g}}{Q^2} <\frac{U_n t_{n,g}}{Q^2} <c_{n,g}+2\cdot10^{-6}. \tag{58}\] Here the centers are interpreted as real numbers, including their \(10^{-6}\) scale. These inequalities in particular put every computed scaled norm below \(4\). By (56), replacing its square by the true trace incurs error less than \((8+4\cdot10^{-7})4\cdot10^{-7}\). Together with (58), this yields \[|Z_n(21/2,g)-c_{n,g}| <2\cdot10^{-6}+(8+4\cdot10^{-7})4\cdot10^{-7} <10^{-5}.\] This proves the \(b=21/2\) table in Lemma 9.

The physical matrix and its rational approximation

In the weight basis, write a word as \(w=(w_0,\ldots,w_{n-1})\), with \(w_i\in\{-1,0,1\}\). The total weight \(\sum_iw_i\) is conserved. In each weight block let \(H\) be the restriction of \(H_n(g)\), and set \[ R=I-\frac{\frac32nI+H}{16},\qquad D=32R. \tag{59}\] By (9), \(D\) is an integer matrix with diagonal \[D_{w,w}=32-3n-2\sum_{i=0}^{n-1}w_{i-1}w_i.\] An allowed opposite unit step on the letters at \(i-1,i\) has entry \(-2\), except that the entry is \(+2\) on the twisted seam \(i=0\). These formulas specify every matrix entry; indices on words are modulo \(n\). The unit step coefficient follows also directly from the two ladder factors \(\sqrt2\) and the transverse prefactor \(1/2\).

Lemma 3 gives \[0\le \frac32nI+H\le\frac52nI,\qquad \operatorname{spec}(R)\subseteq[1-5n/32,1]\subseteq[-7/8,1].\] Thus \(R\) is a self-adjoint contraction. Put \[F=\exp\!\left[-\frac b2\left(\frac32nI+H\right)\right] =\exp[98(R-I)].\] We use the same symbols for the direct sums over all weight blocks. On the full physical space, \[ Z_n(b,g)=e^{x_n}\left\lVert F\right\rVert_F^2,\qquad x_n=bn\left(\frac32-a\right), \tag{60}\] where \(\left\lVert \cdot\right\rVert_F\) denotes the Frobenius norm and the squared norms of the weight blocks are summed.

Define the rational probabilities \[ p_j=\frac{v_j}{\sum_{\ell=0}^Jv_\ell}, \qquad v_j=98^j\frac{J!}{j!}, \qquad 0\le j\le J. \tag{61}\] They are Poisson weights of mean \(98\), conditioned to indices at most \(J\). Since \(\left\lVert R\right\rVert\le1\), discarding the Poisson tail and renormalizing changes the exponential by at most twice the tail probability, by the same coefficient estimate as in Fox–Glynn (Fox and Glynn 1988, Proposition 1): \[\left\|F-\sum_{j=0}^Jp_jR^j\right\| \le 2e^{-98}\sum_{j=281}^{\infty}\frac{98^j}{j!} \le2^{-280}e^{98}<2^{-84}<Q^{-1}.\] For the second inequality, use \(2^j\ge2^{281}\) in the tail and \(e^{-98}\sum_{j\ge0}196^j/j!=e^{98}\); the next uses \(e<4\). The estimate includes the change of normalization.

Integer columns and their error

For a selected unit basis column \(e_r\), start with \(X=0\) and perform, in the order \(j=J,J-1,\ldots,0\), \[ X\ \longleftarrow\ \left\lfloor\frac{DX}{32}\right\rfloor+c_je_r, \qquad c_j=\lfloor Qp_j\rfloor. \tag{62}\] The vector floor is coordinatewise. All quantities in this recursion are integers. The coefficients \(c_j\) vanish for \(j>191\), so those initial zero steps may be omitted without changing \(X\).

To bound rounding, compare with the ideal suffix \[Y_j=Q\sum_{\ell=j}^Jp_\ell R^{\ell-j}e_r,\qquad \left\lVert Y_j\right\rVert_2\le Q.\] In a block of dimension \(d_0\), one vector floor introduces Euclidean error less than \(\sqrt{d_0}\), and one coefficient floor error less than one. The contraction property therefore bounds the accumulated error by \(281(\sqrt{d_0}+1)\). Set \[ d_n=\lfloor\sqrt{3^n}\rfloor+1,\qquad E_n=281(d_n+1)+1,\qquad \mathcal E_n=d_nE_n. \tag{63}\] Including the preceding exponential-approximation error, every computed column satisfies \[ \left\lVert X-QFe_r\right\rVert_2\le E_n. \tag{64}\] The bound charges all \(281\) ideal steps, including the omitted steps whose rational coefficients are nonzero but whose integer floors vanish.

The recursion can be evaluated exactly with signed \(64\)-bit integers. Indeed every intermediate coordinate has absolute value at most \[Q+281(d_n+1)<2Q,\qquad d_n\le730.\] The absolute sum of any row of \(D\) is at most \[|32-3n|+2n+4n\le76.\] The first two terms bound the diagonal, and at most two hops per bond give the last term. Consequently every product and every partial sum in a sparse matrix multiplication has absolute value less than \[76\cdot2Q=5476377146882523136<2^{63}-1.\] Division by \(32\) and addition of \(c_j\le Q\) also fit; even their conservative absolute bound is \[\frac{76\cdot2Q}{32}+Q=207165582859042816<2^{63}-1.\] Starting with \(X=0\), these bounds first establish exactness of the next integer operation; the contraction estimate then establishes the next coordinate bound. Thus the induction does not assume the absence of overflow that it is proving. All final squares and sums below use arbitrary-precision integers.

Representative columns

Only one column from each of the following word orbits is needed. For positive total weight, use the orbit under cyclic shifts and reversal, and give its representative twice the orbit size. This includes the uncomputed negative-weight block. In the zero-weight block, include digit negation in the orbit itself and use its size without an extra factor two. Choose the lexicographically least word as representative. Set cardinalities handle periodic words and all other nontrivial stabilizers. The multiplicities \(\nu_r\) satisfy \[\sum_r\nu_r=3^n.\]

Here are the exact symmetries underlying this reduction. Reversal \(w\mapsto(w_{n-1},\ldots,w_0)\) fixes the seam as an unordered bond, and global negation preserves every Hamiltonian entry. For the twisted chain, a bare shift moves the seam. Left rotation by \(q\) moves it to \((k-1,k)\), where \(k=-q\bmod n\). The diagonal unitary \[G_k|w\rangle=(-1)^{\sum_{j=0}^{k-1}w_j}|w\rangle,\qquad G_0=I,\] flips the hopping signs exactly at the two boundary bonds of this arc. It restores the original seam. The resulting signed shift commutes with \(H\), so exact exponential columns at rotated words have equal norms. For the untwisted chain no sign correction is needed.

Let \(X_r\) be the output of (62) at representative \(r\), and define the integer \[ W_{n,g}=\sum_r\nu_r\sum_w X_r(w)^2. \tag{65}\] Rounding need not preserve the signed symmetries. To compare norms, form abstract direct sums repeating each exact representative column \(\nu_r\) times, and do the same for each computed column. The first direct sum has squared norm \(Q^2\left\lVert F\right\rVert_F^2\), and the second has squared norm \(W_{n,g}\). By (64), \[ \left|\sqrt{W_{n,g}}-Q\left\lVert F\right\rVert_F\right| \le\sqrt{3^n}\,E_n<\mathcal E_n. \tag{66}\] This proves the full trace estimate using representatives, without requiring their rounded columns to transform equivariantly.

Exact totals and rational enclosures

The integer algorithm (61), (62), and (65) gives the following totals. The second column uses no twist; the third uses the \(\pi\) twist.

We finish with a finite rational enclosure, specifying every bound used to turn these integers into the thermal table at \(b=49/4\). Let \[r_{n,g}=\lfloor\sqrt{W_{n,g}}\rfloor,\qquad s_n=\sum_{j=0}^{100}\frac{x_n^j}{j!},\qquad t_n=\frac{x_n^{100}}{100!}.\] Since \(0\le x_n<15\), successive omitted terms have ratio at most \(15/101\), and the tail is at most \(15t_n/86<t_n\). Hence \(s_n\le e^{x_n}\le s_n+t_n\). Equations (60) and (66) give \[ \begin{gathered} L_{n,g}=\frac{s_n\max(0,r_{n,g}-\mathcal E_n)^2}{Q^2},\qquad U_{n,g}=\frac{(s_n+t_n)(r_{n,g}+1+\mathcal E_n)^2}{Q^2},\\ L_{n,g}\le Z_n(b,g)\le U_{n,g}. \end{gathered} \tag{67}\] All quantities in these endpoints are rational or integer, with the integer square root determined by \(r^2\le W<(r+1)^2\). Writing \(C_{n,g}\) for the integer centers in the \(b=49/4\) table of Lemma 9, direct integer and rational evaluation yields \[L_{n,g}>\frac{C_{n,g}-30}{10^6}, \qquad U_{n,g}<\frac{C_{n,g}+30}{10^6} \qquad(4\le n\le12,\ g=I,P).\] Every one of these strict inequalities has margin greater than \(13/10^6\). The displayed algorithm, totals, and enclosures give a reproducible finite verification of all eighteen enclosures, and complete the proof of the thermal table at \(b=49/4\).

Norm and energy contractions

For \(N\ge4\) and three real \(D\)-by-\(D\) matrices \(A_s\), indexed by \(s=-1,0,1\), define a vector in the standard orthonormal spin-one weight basis by \[ \psi_N(s_0,\ldots,s_{N-1}) =\mathop{\mathrm{Tr}}(A_{s_0}\cdots A_{s_{N-1}}). \tag{68}\] This is the periodic matrix-product representation of (Pérez-García et al. 2007, sec. 2.1, Equations (2)–(3)). Write \[\begin{align*} B_{st}&=A_sA_t,\qquad T=\sum_{s=-1}^1 A_s\otimes A_s, \tag{69}\\ O&=\sum_{s,t=-1}^1 st\,B_{st}\otimes B_{st} +\sum_{\substack{s=-1,0\\t=0,1}} \bigl(B_{st}\otimes B_{s+1,t-1} +B_{s+1,t-1}\otimes B_{st}\bigr). \tag{70}\end{align*}\] All traces here are ordinary, unnormalized matrix traces.

One integer matrix triple

Set \(D=26\). Index eight types by \(\alpha=1,\ldots,8\), put \[(\ell_1,\ldots,\ell_8)=(1,1,1,1,3,3,3,5),\] and order the virtual labels \((\alpha,d)\) first by \(\alpha\) and then by \(d=-\ell_\alpha,-\ell_\alpha+2,\ldots,\ell_\alpha\). The matrices \(A_s\) use the diagonal parameters \(D_\alpha\) in Table 1 and the parameters \(P_{\alpha\beta}\) in Table 2, defined only when \(\ell_\beta=\ell_\alpha+2\).

Here are complete entry rules. Consider the row \((\alpha,d)\) and column \((\beta,e)\), put \(\ell=\ell_\alpha\), \(k=\ell_\beta\), and fix \(s\in\{-1,0,1\}\). The entry is zero unless \[ e=d+2s,\qquad |k-\ell|\le2,\qquad (k=\ell\Longrightarrow\alpha=\beta). \tag{71}\] For an allowed entry define \[p=\begin{cases} D_\alpha,& k=\ell,\\ P_{\alpha\beta},& k=\ell+2,\\ -P_{\beta\alpha},& k=\ell-2. \end{cases}\] Put \(r=(d+\ell)/2\) and set the entry equal to \(p f\), where \[ \begin{array}{c|ccc} &s=-1&s=0&s=1\\ \hline k=\ell+2& 2(\ell+1-r)(\ell+2-r)&2(r+1)(\ell-r+1)&(r+1)(r+2)\\ k=\ell& 2(\ell-r+1)&d&-(r+1)\\ k=\ell-2&2&-2&1 \end{array} \tag{72}\] lists the values of \(f\). These entry rules are independent of \(N\).

Finite contractions and variational bounds

We now evaluate the contractions in Lemma 16 for this fixed triple at \(N=72\) and \(N=120\). The support rule (71) makes this a short block calculation. On paired virtual labels \(((\alpha,d),(\beta,e))\), both \(T\) and \(O\) preserve \(d-e\). For \(T\), the two layers shift by the same \(2s\). For \(O\), the two layers have the same total physical weight \(s+t\), also in each off-diagonal term. Thus the difference blocks are exhaustive and have dimensions \[ \begin{array}{c|rrrrrrrrrrr} d-e&-10&-8&-6&-4&-2&0&2&4&6&8&10\\ \hline \text{dimension}&1&8&32&80&136&162&136&80&32&8&1 \end{array}. \tag{73}\] They sum to \(676=26^2\); each block contributes once.

For completeness, the entire integer computation consists of the following operations. Form \(A_s\), \(B_{st}\), \(T\), and \(O\) from Tables 1–2 and Equations (69), (70), (71), and (72). Restrict \(T,O\) to each difference block, obtaining \(T_\delta,O_\delta\). Compute powers by \[P_0=I,\qquad P_{2j}=P_j^2\quad(j\ge1),\qquad P_{2j+1}=P_j^2T_\delta\quad(j\ge0).\] Then accumulate \[ U_N=\sum_\delta\sum_{i,j} (P_{N-2})_{ij}(O_\delta)_{ji}, \qquad V_N=\sum_\delta\sum_{i,j} (P_{N-2})_{ij}(T_\delta^2)_{ji}. \tag{74}\] Only arbitrary-precision integer operations are used. For \(N=72\), the required power is \(70\). For \(N=120\), one may use \(T^{118}=T^{64}T^{32}T^{16}T^4T^2\) and \(T^{120}=T^{64}T^{32}T^{16}T^8\) within each block.

The resulting exact integer comparisons are \[ V_N>0,\qquad -c_NV_N\le10^{12}U_N<(-c_N+1)V_N, \qquad \begin{array}{c|r} N&c_N\\ \hline 72&1401482112601\\ 120&1401482105174 \end{array}. \tag{75}\] The supplied computation files contain the complete matrices and the full integers \(U_N,V_N\), rather than only these shortened intervals. In particular, the files and record the full contractions and positive margins \(-700741V_N-500000U_N\). The common matrices are recorded in . The program , with --length 72 or --length 120, executes the entry rules and block contractions in at the selected length. The integer comparisons in (75) also follow directly from their full recorded integers.

The support rule puts these trial vectors in total weight zero, but the variational bound is for the minimum of the full physical spectrum. Their only symmetry used in the contraction proof is cyclic site translation. The trial inputs hold at every positive inverse temperature, so each finite initialization can use them at its own temperature.