# The periodic spin-one Haldane gap


maketitle
Abstract: 
We prove a uniform positive spectral gap for the antiferromagnetic
spin-one Heisenberg chain on even periodic rings, in the
liminf (even-periodic) formulation of the spin-one Haldane
conjecture.  The proof combines a spatial transfer operator, a
purity bootstrap that doubles length and inverse temperature
together, and two finite, exactly verified initial data sets.


# Introduction

Haldane predicted that antiferromagnetic Heisenberg chains behave
differently for integer and half-odd-integer spin
[cite].  For spin one the
prediction is a positive gap above the ground energy.  We prove this
for the pure bilinear model on even periodic rings, with bounds that
are uniform in the ring length.


Let S^x,S^y,S^zin M_3(C) be self-adjoint spin-one matrices with
[
 [S^alpha,S^beta]=itextstylesum_etaε_{alphabetaeta}S^eta,
  sum_alpha(S^alpha)^2=2I_3 .
]
For even Lge4 put cH_L=(C^3)^{ ⊗  L}, and on it

 H_L=sum_{j=0}^{L-1}sum_{alpha=x,y,z}S_j^alpha S_{j+1}^alpha,
  S_L^alpha=S_0^alpha .

Let E_0(L) be the least energy, E_1(L) the next distinct energy,
gamma_L=E_1(L)-E_0(L), and Delta_1=liminf_{Lto∞,
L even}gamma_L.  Then the ground state of H_L is unique
for every even Lge60, and

 gamma_L&>frac4{105}log(80)/(79) && (Lge60 even),
 
 gamma_L&>(log 20)/(784) && (Lge2304 even),
 

so Delta_1gelog(20)/784>0.  The same conclusions hold for any
simultaneously similar spin-one realization satisfying the same
commutator and Casimir normalizations.


The next distinct energy always exists: in the S^z basis the
all-zero and all-+1 product vectors have energy expectations 0
and L, so H_L is not scalar.  Because H_L is self-adjoint and
non-scalar in finite dimension, its full spectrum has a unique least
eigenvalue and a next distinct one; gamma_L is the difference.

## How the proof works

A shifted partition function Z_L(b)=Tr e^{-b(H_L+aLI)} can be
read in two ways at once.  Physically it is a sum of positive Gibbs
weights, one per physical eigenvector.  Spatially
(Section~sec:transfer) it is a moment Tr(X_b^L) of a
self-adjoint, Hilbert--Schmidt operator X_b; for even L this is a
sum of nonnegative weights, one per spatial eigenvalue.  One object,
two probability distributions.

For a probability distribution (p_i) call sum_i p_i^2 its
purity; purity near one means one weight carries almost all
the mass.  Squaring and renormalizing the weights turns purity defect
1-P into roughly its square (Lemma~lem:purity).  Squaring
physical weights doubles the inverse temperature, while squaring
spatial weights doubles the length.  Exact cancellation identities
couple the two improvements (Lemma~lem:rounded-update), so
starting from two finite purity bounds both defects decay doubly
exponentially as length and inverse temperature double together
(Proposition~prop:bootstrap).  Moment ineualities fill in
every intermediate even length.  Finally, purity of the physical
Gibbs distribution bounds each excited-to-ground weight ratio, hence
the energy gap.

Two finite initializations supply the seeds.  One begins at length
60 and inverse temperature 105/4, the other at length 2304 and
inverse temperature 784 with a smaller defect, giving the stronger
bound.  Both rest on short-chain thermal traces
(Appendices~sec:thermal120--sec:thermal72) and one
explicit family of periodic matrix-product trial vectors
(Appendix~sec:trials).  Every finite step is an exact integer or
rational computation with stated truncation errors; no floating-point
eigenvalue estimate is a premise.

## Historical context

The half-odd-integer side has a rigorous obstruction in the
low-energy bound of Lieb, Schultz and Mattis
cite[Appendix~B, Theorem~2]{LiebSchultzMattis1961}, extended by
Affleck and Lieb [cite]; possible ground-state
degeneracy is part of that picture.  Integer spin admits a unique
gapped state, but this distinction alone does not prove a gap for a
given interaction.

The Affleck--Kennedy--Lieb--Tasaki construction gave a rigorous
gapped spin-one chain with biquadratic interactions
[cite].  Its bond interaction is h+1/3h^2 up
to a scalar, with h=S_j·S_{j+1}, whereas
(eq:main-H) uses 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 point
cite[arXiv v1, Theorems~1 and~3]{Yarotsky2006}; this controls a
neighborhood, not a gapped path to the pure bilinear point.

For the pure chain, White and Huse's DMRG study estimated a bulk gap
of 0.41050(2) and effective spin-one-half end modes
[cite]; neutron scattering found experimental evidence
in the nearly one-dimensional spin-one antiferromagnet
CsNiCl_3 [cite].  Batista and Ortiz proposed a
spin--fermion proof [cite]; its gap step uses a
first-order perturbative reduction without a uniform remainder, so it
does not close the argument.  Tasaki proved a nontrivial
infinite-chain topological index for the model under a gap assumption
for odd open chains with endpoint fields
cite[arXiv v4, Assumption~2 and Theorem~3]{Tasaki2025}; his boundary
hypotheses require an argument separate from the even-periodic
estimates here.

Finite-volume gap criteria include those of Knabe and Nachtergaele
for frustration-free chains [cite], and the
sharper thresholds of Gosset and Mozgunov [cite].
Banerjee, Guo and Chowdhury use semidefinite methods to bound string
correlators
cite[Eqs.~(2),~(13), and Figure~4]{BanerjeeGuoChowdhury2026}, a
different quantity from the uniform spectral gap studied here.
Our finite inputs are thermal traces of small twisted chains and
energies of periodic matrix-product vectors
[cite].  The traces are rigorous enclosures of
matrix-exponential traces; validated matrix exponentiation is
classical [cite], and the
conditioned Poisson polynomial uses the Fox--Glynn estimate
cite[Proposition~1]{FoxGlynn1988}.

The transfer-matrix idea goes back to Suzuki [cite]; our
ordered bond expansion follows Aizenman and Nachtergaele
cite[arXiv v1, Section~2.2]{AizenmanNachtergaele1994}, followed by a
direct operator construction on bond histories.

# The model and its twisted partition functions

A single object will carry the whole proof: a shifted partition
function that is simultaneously a physical Gibbs trace and a spatial
moment.  This section fixes coordinates, twists, and an elementary
energy bound used to control the finite exponential computations.
We use self-adjoint spin-one matrices throughout; the non-self-adjoint
case reduces to this by the next lemma.

[Representation]
Let S^x,S^y,S^zin M_3(C) satisfy
[
 [S^alpha,S^beta]=itextstylesum_gammaε_{alphabetagamma}S^gamma,
  sum_alpha(S^alpha)^2=2I .
]
They are simultaneously similar to the standard self-adjoint spin-one
matrices; if they are already self-adjoint, the similarity is unitary.
The tensor-power similarity preserves the full spectrum of every
periodic H_n.


Proof. 
Put S^pm=S^xpm iS^y.  The assumptions give

 [S^z,S^pm]=pm S^pm,
 S^-S^+=2I-(S^z)^2-S^z, S^+S^-=2I-(S^z)^2+S^z .

Take an S^z-eigenvector v of eigenvalue m with maximal real part.
Then S^+v=0, so m=1 or m=-2.  The case m=-2 would permit
indefinite lowering: a last nonzero lowered vector w has weight
-2-k, while S^+S^-w=0 forces weight -1 or 2, contradicting
finite dimension.  Hence m=1.
The vectors v, S^-v/sqrt2, (S^-)^2v/2 are nonzero by
(eq:ladder-identities) and have distinct weights 1,0,-1; they
form a basis, and there is no weight -2, so they are the whole
standard spin-one module with ladder coefficients sqrt2.  When the
matrices are self-adjoint and norm v=1 the vectors are orthonormal.
Conjugating each tensor factor conjugates every term of H_n, so the
spectrum is unchanged.


In the weight basis |s>, sin{-1,0,1}, the generator is
S^z|s>=s|s> with ladder entries sqrt2.  In the
Cartesian basis indexed by axes, the real matrices
[
 (S^alpha)_{ij}=-iε_{alpha i j}, G_alpha=iS^alpha
]
obey the same relations; each G_alpha is real of operator norm one.
For h=sum_alpha S^alpha ⊗  S^alpha contraction of the epsilon
tensors gives

 -h=sum_alpha G_alpha ⊗  G_alpha,
 h=F-|Ω><Ω|,
 Ω=sum_{i=1}^3|ii>,

where F(u ⊗  v)=v ⊗  u.  Hence h has eigenvalues
-2,-1,1 and hle I.

## Twists and partition functions

Number the sites 0,...,n-1 with nge4.  A proper rotation
gin SO(3) is unitary on C^3.  Define

 H_n(g)=sum_{j=0}^{n-2}h_{j,j+1}+h_{n-1,0}(g),
 h_{n-1,0}(g)=sum_alpha S_{n-1}^alpha,(gS^alpha g^{-1})_0 ,

so only the last bond is conjugated, on its site-0 factor; H_n(g)
is self-adjoint and H_n(I)=H_n.  With the shift

 a=(700741)/(500000),
 Z_n(b,g)=Trexp[-b(H_n(g)+anI)], Z_n(b)=Z_n(b,I),

all Z_n(b,g) are strictly positive, and the scalar shift an will
cancel from every purity ratio and from the final gap.

We need the diagonal rotations
[
 P_x=diag(1,-1,-1),
 P_y=diag(-1,1,-1),
 P_z=diag(-1,-1,1),
]
D_2={I,P_x,P_y,P_z}, and the cyclic rotation
C:e_x|-> e_y|-> e_z|-> e_x; write P=P_x.  By a global
cyclic rotation the three nonidentity elements of D_2 have equal
partition functions.  After a global rotation and a change to the
weight basis, P and C act as exp(-itheta S^z) with
theta=pi and theta=2pi/3.

For later use with explicit matrices, a word x=(x_0,...,x_{n-1})
in the weight basis has diagonal entry sum_j x_{j-1}x_j (indices
mod n); each allowed step changing two neighboring letters by
opposite units contributes 1, and a seam step from column y to row
x contributes the phase

 exp!(itheta(y_0-x_0)),

since gS^pm g^{-1}=e^{mp itheta}S^pm.  These formulas tie the
finite matrices below to the physical H_n(g) in
(eq:twisted-hamiltonian).

## A uniform energy bound

[Energy bounds]
For every nge4 and gin SO(3), -(3n)/(2)Ile H_n(g)le nI.


Proof. 
The upper bound is hle I plus unitary conjugation of the seam.  For
the lower bound consider T=-h_{12}-h_{23} on three open sites in
Cartesian coordinates.  By (eq:cartesian-bond) an uneual pair
swaps with coefficient -1, an equal pair changes to either other equal
pair with coefficient 1, and there are no diagonal terms.  Give each
three-letter word a positive weight q according to its equality
pattern:

{c|c|c}
word type & q & absolute weighted row sum 
aaa & 4 & 3+3+3+3=12
aab or abb & 3 & 4+3+2=9
aba & 2 & 3+3=6
abc & 2 & 2+2=4


Every row sum is at most 3q.  With D the diagonal of these weights,
D^{-1}TD has absolute row sums at most three, hence spectral radius
at most three; since T is self-adjoint and similar to this matrix,
h_{12}+h_{23}ge-3I.

An open triad containing the seam is unitarily equivalent to an
untwisted triad (delete the seam bond and rotate all sites on one side,
using rotation invariance of the internal interaction), so the bound
holds for every consecutive pair of bonds of H_n(g).  Summing the n
triad ineualities counts each bond twice and gives
H_n(g)ge-(3n)/(2)I.


The trial inputs of Appendix~sec:trials give E_0(n)+an<0 at
n=72,120, hence Z_n(b)>1 for every b>0; those lower bounds are
paired with finite thermal upper bounds in Section~sec:initialization.

# Partition functions as spatial moments

We now construct a self-adjoint spatial operator whose Lth moment is
the physical partition function (eq:shift).  This is what turns
even spatial powers into positive weights.  The operator itself need
not be positive.

Let G SO(3) be the group of proper signed permutation
matrices (it contains D_2 and C).  For ginG write
[
 g e_alpha=ε_g(alpha)e_{sigma_g(alpha)},
  ε_g(alpha)in{-1,1}.
]
Rotation covariance of the Cartesian matrices gives

 gG_alpha g^{-1}=ε_g(alpha)G_{sigma_g(alpha)} .


[Spatial transfer]
For each b>0 there are a Hilbert space K_b, a compact
self-adjoint operator X_b, and a unitary representation
g|-> U_g of G commuting with X_b, such that

 Z_n(b,g)=Tr(X_b^nU_g)(nge4, ginG).

Both X_b and the representation commute with complex conjugation, and
X_b is Hilbert--Schmidt, so X_b^k is trace class for every integer
kge2.  If lambda_i(b) are the nonzero eigenvalues, counted with
multiplicity, then

 sum_i|lambda_i(b)|^k<∞ (kge2),
 Z_n(b)=sum_ilambda_i(b)^n (nge4).



Proof. 
Histories and kernel.  A bond history is a finite ordered list
[
 x=((t_1,alpha_1),...,(t_k,alpha_k)),
 0<t_1<...<t_k<b, alpha_jin{x,y,z},
]
including the empty list.  Let Ω_b be the disjoint union of
these labeled simplexes, with Lebesgue measure on each and unit mass on
the empty history.  The measure is finite:

 mu(Ω_b)=sum_{kge0}(3^kb^k)/(k!)=e^{3b},
  K_b=L^2(Ω_b,K_b=L^2(Ω_b,mu;C).

For two histories x,y, merge their labeled times increasingly and let
k_b(x,y) be the trace of the resulting time-ordered product of the
G_alpha (empty pair: the identity, trace three; time collisions: a
null event, set kernel zero).  Since norm{G_alpha}=1 the kernel is
real with |k_b(x,y)|le3, and exchanging x,y keeps the merged
product unchanged, so k_b(y,x)=k_b(x,y).  The integral operator
K_b is thus self-adjoint and Hilbert--Schmidt with
norm{K_b}_{HS}^2le9e^{6b}.  Set X_b=e^{-ab}K_b; compact
self-adjoint spectral theory gives real square-summable eigenvalues,
and boundedness gives sum_i|lambda_i|^k<∞ for kge2.

Rotations.  The permutation sigma_g acts on labels preserving
times and measure.  Put s_g(x)=prod_{alphain x}ε_g(alpha)
and (U_g f)(x)=s_g(sigma_g^{-1}x)f(sigma_g^{-1}x).  The cocycle rule
s_{gh}(x)=s_g(sigma_hx)s_h(x) shows these real unitaries form a
representation.  Rotating every matrix inside a site trace, via
(eq:rotation-covariance), gives
k_b(sigma_gx,sigma_gy)=s_g(x)s_g(y)k_b(x,y), so U_g commutes with
K_b, X_b; the kernel of K_bU_g is

 (K_bU_g)(x,y)=k_b(x,sigma_gy)s_g(y).


The cycle trace.  For nge2 both K_b^{n-1} and K_bU_g are
Hilbert--Schmidt (the iterated kernel of K_b^{n-1} is bounded because
k_b is bounded and mu finite).  The Hilbert--Schmidt product trace
identity Tr(AB)=int a(x,y)b(y,x),dmu x,dmu y and
(eq:twisted-kernel) give

 Tr(K_b^nU_g)=int_{Ω_b^n}(prod_{j=0}^{n-2}k_b(x_j,x_{j+1}))
 k_b(x_{n-1},sigma_gx_0)s_g(x_0),dmu^n ,

with all iterated integrals absolutely integrable, so no diagonal
formula for K_b is needed.

Identification.  Expand exp[-bH_n(g)] in its ordered-time
series.  By (eq:cartesian-bond) each event picks a bond and one of
three labels and inserts the corresponding two G factors; at the seam
the site-0 factor is rotated.  At total degree k there are (3n)^k
choices each of tensor norm at most one, so absolute convergence holds:

 3^nsum_{kge0}((3nb)^k)/(k!)=3^ne^{3nb}.

We may regroup by bond; interleavings of bond histories partition the
global time simplex up to null collisions, and at each site the tensor
trace is the kernel on the two incident histories in their original
chronological order.  For an explicit seam check, call its history
e_0 and the others e_1,...,e_{n-1} around the ring.  The physical
integrand is
[
 s_g(e_0)k_b(sigma_ge_0,e_1)k_b(e_1,e_2)... k_b(e_{n-1},e_0),
]
which equals the right side of (eq:cycle-integral) after setting
x_0=e_0,x_1=e_{n-1},...,x_{n-1}=e_1 (spatial enumeration reversed,
chronological products untouched).  Thus
Tr e^{-bH_n(g)}=Tr(K_b^nU_g), and multiplying by e^{-abn} gives
(eq:spatial-trace); the compact trace formula gives
(eq:spatial-moments).


For even nge4, (eq:spatial-trace) supplies the probabilities

 p_i=(|lambda_i(b)|^n)/(Z_n(b)),
 sum_i p_i=1,
 sum_i p_i^2=frac{Z_{2n}(b)}{Z_n(b)^2}.

Odd moments are also useful for polynomial estimates, but evenness is
essential here.  The spatial eigenvalues are distinct from the physical
energies of H_n.

## Rotation sectors

Small twisted partition functions split the spatial spectrum into a few
lists with known multiplicities; this later isolates a dominant weight.

[Sector moments]
Fix b>0 and let X_b,U_g be as in Proposition~prop:transfer.
There are real eigenvalue lists (with multiplicity) whose integer
moments I_k,O_k,N_k converge absolutely for kge2, with

 Z_n(b)&=I_n+2O_n+3N_n,
 Z_n(b,P)&=I_n+2O_n-N_n,
 Z_n(b,C)&=I_n-O_n (nge4).

The I list is the restriction to the joint invariant subspace of
D_2 and C; the O list occurs twice and the N list three times
in the full spatial list.  All three moments are nonnegative at even
order, and J_n=I_n+2O_n is the moment on the D_2-invariant subspace.


Proof. 
The four D_2 characters define orthogonal projections
Pi_chi=frac14sum_{gin D_2}chi(g)U_g commuting with X_b and
summing to the identity.  Conjugation by U_C permutes the three
nontrivial character spaces cyclically, so their restrictions are
unitarily equivalent; write N_k for the common moment.  In the
trivial-character space, split by the eigenvalues 1,omega,baromega
of U_C (omega=e^{2pi i/3}); these reduce X_b, complex
conjugation exchanges the last two spaces and preserves each real
eigenvalue of X_b with multiplicity, so their common moment is O_k
and the first space gives I_k.  Absolute convergence is
(eq:spatial-moments).

The decomposition gives (eq:sector-total).  On the three
nontrivial spaces a fixed U_P has signs +1,-1,-1 and is the identity
on the trivial space, giving (eq:sector-pi).  The operator U_C
permutes the nontrivial spaces (zero contribution) and has coefficients
1,omega,baromega on the remaining three with
omega+baromega=-1, giving (eq:sector-cyclic).  The J_n
identity is the decomposition of the trivial space, with nonnegativity
at even order.


The invariant I space need not be one-dimensional; the point is that
every weight outside it occurs at least twice in the full list, which
identifies a unique largest weight without discarding any sector.

# From two purities to a uniform gap

The spatial and physical descriptions of the same partition
function give two ways to concentrate its weights.  This section turns
two initial concentration estimates into a gap at every larger even
length; the initial estimates are supplied in
Section~sec:initialization.

By Proposition~prop:transfer, each partition function is both a
physical Gibbs trace (counting every physical eigenvector with
multiplicity) and, at even lengths, a nonnegative spatial moment
(counting absolute powers of every spatial eigenvalue).  Define, for
even nge4,

 S(n,b)=frac{Z_{2n}(b)}{Z_n(b)^2},
 T(n,b)=(Z_n(2b))/(Z_n(b)^2).

The shift aL cancels from both ratios.  First, the ineuality behind
their improvement under squaring.

[Squaring probabilities]
Let (w_i) be a finite or countable family of nonnegative numbers
with sum_iw_i=1 and purity P=sum_iw_i^2in(0,1].  The normalized
squares w_i^2/P have purity P' with

 1-P'le f(1-P), f(u)=(u^2)/(2(1-u)^2)  (0le u<1),

and f is increasing.  Consequently

 0<S(n,b),T(n,b)le1,
 1-S(2n,b)le f(1-S(n,b)),
 1-T(n,2b)le f(1-T(n,b)).



Proof. 
Some weight is positive, so P>0, and w_i^2le w_i gives Ple1.
Nonnegativity allows rearrangement of all countable sums, so
[
 1-P'=frac{2sum_{i<j}w_i^2w_j^2}{P^2}
 lefrac{2(sum_{i<j}w_iw_j)^2}{P^2}
 =((1-P)^2)/(2P^2),
]
and f'(u)=u/(1-u)^3ge0.  For S(n,b) use the spatial
probabilities |lambda_i(b)|^n/Z_n(b) of
(eq:spatial-probabilities): squaring them changes n to 2n.
For T(n,b) list all physical eigenvalues ε_k(n) with
multiplicity and use

 w_k(n,b)=frac{e^{-b(ε_k(n)+an)}}{Z_n(b)} .

Their purity is T(n,b), and squaring them changes b to 2b.


## The coupled update

The two purities constrain one another because they share partition
functions; direct cancellation gives

 S(n,2b)&=S(n,b)^2(T(2n,b))/(T(n,b)^2),
   
 T(4n,b)&=T(2n,b)^2(S(2n,2b))/(S(2n,b)^2).
   

Both denominators are at most one, and the following form lets the two
initial bounds differ and be rounded upward.

[Coupled update]
Suppose nge4 even, b>0, and p,qin[0,1) with
1-S(n,b)le p, 1-T(2n,b)le q.  If p_+,q_+in[0,1) obey

 p_+ge f(1-(1-p)^2(1-q)),
 q_+ge f(1-(1-q)^2(1-p_+)),

then 1-S(2n,2b)le p_+ and 1-T(4n,2b)le q_+.


Proof. 
Equation (eq:space-time-cancel) gives
S(n,2b)ge(1-p)^2(1-q); spatial squaring in Lemma~lem:purity
and monotonicity of f prove the first bound.  Next
(eq:time-space-cancel) gives T(4n,b)ge(1-q)^2(1-p_+), and
temporal squaring proves the second.  Every input of f lies in [0,1)
because the product subtracted from one is positive and at most one.


In words: round the first output up, feed it into the second, round
that up.  The pair based at (n,b) becomes a pair based at (2n,2b)
whose physical member sits at length 4n; neither bound has to
decrease.

## Quadratic decay and intermediate lengths

Once both defects are small one common bound suffices.  Set

 G(u)=frac12((1-u)^{-1}+(1-u)^{-2}+(1-u)^{-3})^2,

increasing on [0,1), with

 f(1-(1-u)^3)=G(u)u^2 .


[Uniform gap from two initial purities]

For every even Lge4 let H_L be a non-scalar finite-dimensional
self-adjoint Hamiltonian, and suppose that ainR and, for each
b>0, a real finite or countable list (lambda_i(b)) satisfy
[
 sum_i|lambda_i(b)|^4<∞,
 Z_L(b):=Tr e^{-b(H_L+aLI)}=sum_i|lambda_i(b)|^L
 (Lge4 even).
]
Define S,T by (eq:purities) and G by
(eq:bootstrap-coefficient).  Let n_0ge4 even, b_0>0, and let
u_0,C satisfy

 0<u_0<1/6, G(u_0)le C, r:=Cu_0<1, C(1-3u_0)>3,

with seeds

 S(n_0,b_0)ge1-u_0, T(2n_0,b_0)ge1-u_0 .

For jge0 set

 n_j=2^jn_0, b_j=2^jb_0, u_j=C^{-1}r^{2^j}.

Then

 S(n_j,b_j)ge1-u_j, T(2n_j,b_j)ge1-u_j,

and for every even Lge n_0 the physical ground state is unique with
gap

 gamma_L>(log(1/r))/(b_0).



Proof. 
Doubling.  By definition u_{j+1}=Cu_j^2, 0<u_{j+1}<u_jle
u_0.  Suppose both defects at (n_j,b_j) are at most u=u_j.  The
first bound in Lemma~lem:rounded-update is at most
[
 f(1-(1-u)^3)=G(u)u^2le Cu^2=u_{j+1}le u .
]
Use p_+=u_{j+1} in the second update; its input is at most
1-(1-u)^3, so its output is also at most Cu^2=u_{j+1}.  Induction
gives (eq:dyadic-purities).

Intermediate even lengths.  For a nonnegative list (a_i) with
finite positive pth-power sum and qge p>0, normalization gives
sum a_i^qle(sum a_i^p)^{q/p} (the normalized pth powers are
probabilities, whose q/p powers sum to at most one).  Applied at the
two temperatures, whenever nle Lle2n with both even,
[
 Z_L(b)le Z_n(b)^{L/n},
 Z_L(2b)ge Z_{2n}(2b)^{L/(2n)},
]
which combine to

 T(L,b)ge[T(2n,b)S(n,b)^2]^{L/(2n)} .

The bracket lies in (0,1] and the exponent in [1/2,1].  For
n=n_j, b=b_j:

 T(L,b_j)ge(1-u_j)^3ge1-3u_j>1/2
 (n_jle Lle2n_j, L even).


The physical spectrum.  The largest weight in
(eq:gibbs-weights) is at least its purity, since
sum_kw_k^2le(max_kw_k)sum_kw_k, hence >1/2.  These weights count
all physical eigenvectors, so two ground-state eigenvectors would give
two equal maximal weights, impossible: the ground state is unique.
Since H_L is non-scalar and self-adjoint in finite dimension, a next
distinct energy exists.  Let w_0 be the ground weight and w_1 the
weight of an eigenvector at that next energy.  Then

 e^{-b_jgamma_L}
 =(w_1)/(w_0)
 le(1-w_0)/(w_0)
 le(3u_j)/(1-3u_j)
 le(3)/(C(1-3u_0))r^{2^j}
 <r^{2^j},
 

using the shift cancellation and (eq:bootstrap-hypotheses).
Taking logarithms and b_j=2^jb_0 gives (eq:bootstrap-gap) on
n_jle Lle2n_j; the dyadic intervals cover every even Lge n_0.


## The two parameter choices

We will prove the seeds (eq:bootstrap-seeds) for

{c|c|c|c|c}
 n_0&b_0&u_0&C&r=Cu_0 
 60&105/4&1/8&79/10&79/80
 2304&784&1/100&5&1/20


The scalar hypotheses are checked directly.  For the first row,
G(1/8)=913952/117649<79/10 and
3/[C(1-3u_0)]=48/79<1.  For the second, 1-(1-u)^3le3u gives for
0<ule1/100
[
 G(u)lefrac9{2(1-3u)^2}le(45000)/(9409)<5,
 frac3{C(1-3u_0)}=(60)/(97)<1 .
]
So the two rows give the bounds (4/105)log(80/79) and (log20)/784
of (eq:bootstrap-gap), once their seeds hold.  The second row will
come from six applications of Lemma~lem:rounded-update to bounds
based at (36,49/4).  Everything left is finite: establish the two seed
ineualities without losing the margins these updates need.

# Two finite initializations

The bootstrap needs a spatial purity at length n and a physical
purity at length 2n, at the same inverse temperature.  We build two
such seed pairs.  The first starts at length 60; the second reaches
length 2304 and gives the stronger eventual gap.  Each seed rests on
a few short-chain partition functions and one periodic trial vector.

## Thermal data and moment bounds

Write J_n=I_n+2O_n for the D_2-invariant moment, I_n for the
moment also fixed by C, and N_n for one nontrivial D_2 sector
(Lemma~lem:sectors).  Since each O_n occurs twice and each
N_n three times,


 J_n&=frac{Z_n+3Z_n(P)}4,& N_n&=frac{Z_n-Z_n(P)}4,
 I_n&=(Z_n+3Z_n(P)+8Z_n(C))/(12),&
 O_n&=(Z_n+3Z_n(P)-4Z_n(C))/(12),


all at one common inverse temperature.

[Finite thermal inputs]
For a=700741/500000 the tables below give centers of the shifted
Z_n(b,g) in units 10^{-6}, with absolute error below 10^{-5} at
b=21/2 and below 30·10^{-6} at b=49/4:
[
{c|rrr}
 &multicolumn{3}{c}{b=21/2}
 n&g=1&g=P&g=C 
 6&8945303&346734&939987
 8&3741281&567167&984108
 10&2327547&721090&1002613


{c|rr}
 &multicolumn{2}{c}{b=49/4}
 n&g=1&g=P 
 4&124885379&65757
 5&2925&2960742
 6&12870614&286599
 7&54069&1920414
 8&4639283&510140
 9&195455&1514846
 10&2654823&675419
 11&377629&1314544
 12&1900547&787405

]


Appendices~sec:thermal120 and~sec:thermal72 prove these
tables by rational enclosures of the physical matrix exponentials with
explicit error bounds; they are rigorous, not numerical estimates.  The
other finite input is Lemma~lem:trial-inputs:

 E_0(72)<-72a, E_0(120)<-120a,

so Z_{72}(b)>1 and Z_{120}(b)>1 for every b>0 (one Boltzmann
factor already exceeds one).

Two elementary lemmas convert the tables into spectral bounds.

[Capped masses]
Let (t_i) be finite or countable with 0le t_ile h and
sum_it_ile m, where h>0, mge0.  For rge1 put

 C(m,h,r)=qh^r+(m-qh)^r,C(m,h,r)=qh^r+(m-qh)^r, q=floor( m/h).

Then sum_it_i^rleC(m,h,r);C(m,h,r); without the cap,
sum_it_i^rle m^r.


Proof. 
For a finite family at fixed total mass, move mass from the smaller of
two coordinates in (0,h) to the larger until one reaches 0 or h;
convexity of x|-> x^r shows the sum of rth powers does not
decrease.  Repetition leaves at most one coordinate in (0,h), giving
(eq:capped-function) at the actual mass; the right side is
continuous and nondecreasing in the mass, so bounded by its value at
m.  Take increasing unions of finite subfamilies for the countable
case.  The uncapped bound follows from t_i^rle t_i(sum_jt_j)^{r-1}.


[Polynomial filter]
Let εge0, v>0, and let (lambda_i) be a real eigenvalue
list with moments M_j=sum_ilambda_i^j converging absolutely, and
|M_j-c_j|leε for the exponents occurring in a polynomial
F(x)=sum_jf_jx^jge0 on R.  If
[
 B=sum_jf_jc_j+εsum_j|f_j|
 quadandquad F(x)>B for |x|ge v,
]
then every |lambda_i|<v.


Proof. 
Absolute convergence gives sum_iF(lambda_i)le B; nonnegativity
means a single |lambda_i|ge v would contradict this.


All moments converge by Proposition~prop:transfer, and the
coefficient sums in (eq:sector-inversion) are at most one in
absolute value, so the thermal error bound also bounds each sector
moment error (odd moments included, with signs kept in the filter).

## The initialization at length sixty


At inverse temperature b=105/4, S(60,b)>7/8 and T(120,b)>7/8.


Proof. 
Work at b_0=21/2.  From the b_0=21/2 table and
(eq:sector-inversion) the tenth moments obey
[
 m_I=(1042655)/(10^6), m_O=(40041)/(10^6),
 m_N=(401625)/(10^6),
]
with margins 19/(12·10^6), 7/(12·10^6),
3/(4·10^6) over centers plus 10^{-5}.  Apply
Lemma~lem:polynomial-filter to
[
 F_I(x)=x^6(-2+10x^2)^2, F_N(x)=x^6(-25+100x^2)^2 .
]
For a filter x^6(c_0+c_1x^2)^2 the summed upper bound is the centered
combination of moments 6,8,10 with coefficients c_0^2,2c_0c_1,c_1^2
plus 10^{-5}(|c_0|+|c_1|)^2.  Rational evaluation gives
[
 |lambda|<U:=(100245)/(100000) (I sector),
 |lambda|<V:=(88643)/(100000) (N sector),
]
since the filter at the endpoint exceeds its bound by more than
0.0301 (I) and 0.3463 (N), and each filter increases with |x|
beyond its endpoint (c_1>0, c_0+c_1x^2>0 there).

The ineualities U^{10}<m_I<2U^{10} and V^{10}<m_N<2V^{10} allow the
two-piece (eq:capped-function).  For k=30,120 set
[
 A_k=U^k+(m_I-U^{10})^{k/10}, B_k=m_O^{k/10},
 C_k=V^k+(m_N-V^{10})^{k/10},
]
bounding I_k,O_k,N_k.  These are even moments, hence nonnegative, so

 Z_k(b_0)le A_k+2B_k+3C_k,
 Z_k(b_0,P),Z_k(b_0,C)le A_k+2B_k .

Put m=A_{30}+2B_{30}+3C_{30}, p=A_{30}+2B_{30},
q=A_{120}+2B_{120}+3C_{120}, and
[
 D=(1450)/(1000), E=(1202)/(1000),
 tau=(1225)/(1000),quadtau_0=(1264)/(1000),
 R_q=(684)/(10000).
]
The scalar estimates are

 m^5<D^2, p^5<E^2, qge1, (q-1)^5<R_q^2,
 (D+11E)/(12)letau, (D+3E)/(4)letau_0,

with positive margins exceeding 0.001068, 0.000433,
0.0000472 in the first, second and fourth strict ineualities; all
numbers are rational and stated above.

Raise the inverse temperature to b=(5/2)b_0.  For positive Boltzmann
factors z_i, sum_i z_i^{5/2}le(sum_iz_i)^{5/2}, so
(eq:short-moment-upper)--(eq:seed60-scalars) give
[
 Z_{30}(b)le D, Z_{30}(b,P),Z_{30}(b,C)le E ;
]
by (eq:sector-inversion) the I and J portions of the thirtieth
spatial moment are at most tau and tau_0.

Let w_i=|lambda_i(X_b)|^{30}, so sum_iw_ile D, while
(eq:trial-thermal-consequence) gives sum_iw_i^4=Z_{120}(b)>1.
Since D(0.88)^3<1, some w_0>0.88 exists, and it is unique because
2(0.88)>D.  It lies in I: an O weight occurs at least twice and an
N weight at least three times (counting multiplicities, including
equal moduli of opposite-sign spatial eigenvalues).

To sharpen w_0, suppose 0le tle0.998 and w_0ge t.  Let x be
the remaining I mass, y the full O-pair mass, z the full N-triple
mass; their squared weights sum to at most x^2+y^2/2+z^2/3 with
[
 0le xletau-t, 0le y,
 x+yletau_0-t, zle D-t-x-y .
]
The divisors use identical repeated lists.  Since D>tau_0 the last
bound is nonnegative on the whole polygon, and the convex quadratic
x^2+y^2/2+(D-t-x-y)^2/3 attains its maximum at a vertex:

 R(t)=max_{(x,y)inV_t}
 (x^2+(y^2)/(2)+((D-t-x-y)^2)/(3)),
 V_t={(0,0),(tau-t,0),(tau-t,tau_0-tau),(0,tau_0-t)}.

Thus the remaining squared weights sum to at most R(t), their fourth
powers to at most R(t)^2.  Exact evaluation gives
[
{c|ccc}
 t&0.88&0.99&0.998 
 R(t)&0.1359&0.0721&0.068404

]
and
[
 1-0.99^4-R(0.88)^2=0.02093518,
 1-0.998^4-R(0.99)^2=0.002777621984.
]
If w_0le0.99 the first would contradict sum_iw_i^4>1, and the
second then excludes w_0le0.998.  Keeping the dominant contribution
in the fourth moment,
[
 S(60,b)=(sum_iw_i^4)/((sum_iw_i^2)^2)
 ge(1+(R(0.998))/(0.998^2))^{-2}>0.8756>frac78 .
]

For the physical purity at length 120, at temperature b_0: the
largest Boltzmann factor z_0 exceeds one (trial bound) while the sum
of all factors is at most q.  After raising to the power 5/2 the
ratio of remaining mass to z_0^{5/2} is at most (q-1)^{5/2}<R_q,
so
[
 T(120,b)ge(1+R_q)^{-2}>0.8760>frac78 .
]
This last step concerns physical Boltzmann factors, separate from the
spatial weights used for S.


## The initialization giving the stronger gap


At inverse temperature b=784,
[
 1-S(2304,b)le(84513)/(10^7)<frac1{100},
 1-T(4608,b)le(27493)/(5·10^6)<frac1{100}.
]


Proof. 
We isolate one large J eigenvalue, bound the rest, get seeds at
length 36, then apply six rounded updates to reach 2304.

Start at b_0=49/4; the b_0=49/4 table gives centers for J_n,N_n
via (eq:sector-inversion) with error at most ε=30·10^{-6}.
Use
[
 F_J(x)=x^4(7+17x-280x^2-185x^3+1000x^4)^2,
 F_N(x)=x^4(12-394x^2+1000x^4)^2,
]
of degree 12, so the table supplies every needed moment.  The summed
upper bounds of Lemma~lem:polynomial-filter are
[
 B_J=318604.54848175, B_N=48169.880699 .
]
Set v_J=100139/100000, v_N=872/1000.  Direct comparison gives
[
 F_J(v_J)-B_J>180, F_J(-v_J)-B_J>496888,
 F_N(pm v_N)-B_N>654 .
]
To cover the whole exterior rays, write Q(x)=7+17x-280x^2-185x^3+1000x^4;
for yge0,
[
 
 Q(1+y)&=559+2902y+5165y^2+3815y^3+1000y^4,
 Q(-1-y)&=895+3978y+6275y^2+4185y^3+1000y^4.
 
]
both positive and increasing.  Likewise 12-394r^2+1000r^4 is positive
and increasing for rge v_N (value at v_N positive, derivative
4r(1000r^2-197)>0).  Hence every J eigenvalue has modulus below
v_J and every N eigenvalue below v_N.

The twelfth-moment upper bounds are m_J=1.0657205, m_N=0.2783155.
Put ell=999/1000 and, for k=36,72,

 B(k)=3C(m_N,v_N^C(m_N,v_N^{12},k/12),
 R_A(k)=B(k)+(m_J-ell^{12})^{k/12}

(with m_J>ell^{12}).  If all J eigenvalues had modulus at most
ell, Lemma~lem:cappedmass would give
Z_{72}(b_0)leC(m_J,ell^{12},6)+B(72)<1, with strict margin
above 0.0693; the trial ineuality contradicts this.  So some
lambda_0 in J has |lambda_0|>ell; removing it leaves twelfth
moment at most m_J-ell^{12} in J, and with the three N sectors
R_A(k) bounds the remaining full kth moment.  Thus


 1-S(36,b_0)&le p_0:=1-(1+frac{R_A(36)}{ell^{36}})^{-2},
 1-T(72,b_0)&le q_0:=1-(v_J^{72}+R_A(72))^{-2}.


The first retains lambda_0^{72}; the second uses Z_{72}(2b_0)>1 and
Z_{72}(b_0)le v_J^{72}+R_A(72).  Numerically p_0<0.047916,
q_0<0.181521, but the updates use their exact rational definitions.

Apply Lemma~lem:rounded-update six times, rounding each new defect
up to a multiple of 10^{-7} and using the rounded spatial defect in
the physical update.  With f(u)=u^2/[2(1-u)^2] and
lceil xrceil_7=10^{-7}lceil10^7xrceil:
[
 p_{j+1}=bigllceil f(1-(1-p_j)^2(1-q_j))bigrrceil_7,
 
 q_{j+1}=bigllceil f(1-(1-q_j)^2(1-p_{j+1}))bigrrceil_7 .
]
Exact rational evaluation gives
[
{c|cc}
 j&p_j&q_j 
 1&151249/2500000&433453/2500000
 2&343303/5000000&1632381/10000000
 3&44601/625000&361773/2500000
 4&632939/10000000&1055161/10000000
 5&375793/10000000&445941/10000000
 6&84513/10000000&27493/5000000

]
Each update lies in [0,1), where f increases, so upward rounding
preserves the bounds.  Length and inverse temperature have multiplied by
2^6: 36·64=2304 and (49/4)·64=784.  The last row proves
the proposition.


## A periodic interface for boundary estimates

[Periodic inputs for boundary estimates]

Let X_b be the transfer operator of Proposition~prop:transfer
with a=700741/500000, and put
[
 
 b_0=49/4, b_1=49/2, ell=999/1000,
 U=100139/100000, V=872/1000, m_J=1.0657205, m_N=.2783155 .
 
]
At b=b_0 the full spatial list is the D_2-invariant list J plus
three identical lists N.  Every J eigenvalue has modulus below U
and the sum of its twelfth powers is at most m_J; every N eigenvalue
has modulus below V and each N list's twelfth powers sum to at most
m_N; some J eigenvalue has modulus above ell.  Moreover
1-S(72,b_1)le151249/2500000.  For n=72,120, E_0(n)+na<0, so a
physical ground Boltzmann factor of Z_n(b) exceeds one for every
b>0.


Proof. 
The decomposition is Lemma~lem:sectors with J=I+2O; the filters,
twelfth-moment bounds and dominant eigenvalue are proved at the start of
Proposition~prop:seed2304.  The first row of the rounded update
table gives the display at (72,49/2); the variational assertion is
Lemma~lem:trial-inputs.


# Proof of the gap theorem

The two initializations feed the same purity theorem.  The finite inputs
they use are proved in Appendices~sec:thermal120--sec:trials;
those proofs are part of the argument.

Proof. [Proof of Theorem~thm:main]
Proposition~prop:transfer identifies each partition function with
a moment of a self-adjoint spatial transfer operator, so S and T of
Section~sec:bootstrap are well defined.  The H_L are self-adjoint
and non-scalar, so the bootstrap criterion applies.

Proposition~prop:seed60 gives
S(60,105/4)>7/8 and T(120,105/4)>7/8.  Apply
Proposition~prop:bootstrap with
(n_0,b_0,u_0,C)=(60,105/4,1/8,79/10); its parameter ineualities hold
by Section~sec:bootstrap, and Cu_0=79/80.  Hence the ground
state is unique and gamma_L>frac4{105}log(80/79) for every even
Lge60, proving (eq:gap60).

Proposition~prop:seed2304 gives both defects at most 1/100 at
the pair with (n_0,b_0)=(2304,784).  Apply the same theorem with
u_0=1/100, C=5; since Cu_0=1/20,
[
 gamma_L>(log20)/(784)(Lge2304 even),
]
proving (eq:gap2304).  The limit inferior over even lengths gives
Delta_1gelog(20)/784>0.  Finally, Lemma~lem:representation
carries every spectral conclusion to any simultaneously similar spin-one
realization, via the tensor-power similarity.


The bounds become uniform on all even lengths as follows: the
second bound covers Lge2304, and the finitely many even L with
4le Lle2302 are finite-dimensional self-adjoint Hamiltonians, each
with a positive gap above its full ground sector.  This yields
inf_{Lge4,,L even}gamma_L>0, though it gives neither a
numerical value nor simplicity at those small lengths.

## An infinite-volume consequence

The finite-volume estimate also controls local excitations in a
thermodynamic limit, using the limiting argument of
cite[arXiv v4, End Matter, Definition~11 and Theorem~12]{Tasaki2025}.
A local operator on the spin chain Z acts on finitely many sites; a
state omega is a linear functional on local operators with
omega(I)=1, omega(A^*A)ge0.  Put
[H,A]=sum_{jinZ}[S_j·S_{j+1},A], only finitely
many terms nonzero.


For the self-adjoint Hamiltonians (eq:main-H), the normalized
ground states of the even periodic rings have a subsequence, as
Lto∞, converging on every local operator to a state omega;
each such limit obeys

 omega(A^*[H,A])ge(log20)/(784)(omega(A^*A)-|omega(A)|^2)

for every local operator A.


Proof. 
Place the even ring of length L on {-L/2,...,L/2-1} with its
closing bond at the endpoints.  Compactness of the density matrices on
each finite interval and a diagonal subsequence give convergence on all
local operators; the compatible limits form a positive normalized
functional omega.  For Lge2304, Theorem~thm:main gives a
unique ground state psi_L and gap above delta=log(20)/784.
Writing omega_L(A)=<psi_L,Apsi_L>, the spectral
ineuality on the orthogonal complement of psi_L gives
[
 omega_L(A^*[H_L,A])gedelta(omega_L(A^*A)-|omega_L(A)|^2).
]
For fixed local A and large L, the closing bond commutes with A and
[H_L,A]=[H,A]; passing to the local limit proves the claim.


In particular omega is a ground state (no local perturbation lowers
its energy).  For omega(A)=0, (eq:local-limit-gap) is the
locally unique gapped property of
cite[arXiv v4, Definition~11]{Tasaki2025}, with gap at least
log(20)/784.  The argument does not identify limits of different
subsequences, nor uniqueness among all infinite-volume ground states.

appendix
# Thermal calculation at inverse temperature 21/2

We prove the b=21/2 table of Lemma~lem:thermal-inputs.  The
calculation uses the physical Hamiltonians at n=6,8,10 with twists
1,P,C; all arithmetic is integral or rational, complex numbers being
written in Z[omega] with omega=e^{2pi i/3}.

## Finite matrices and their columns

Fix b=21/2 and one length, write H=H_n(g).  The three twists have
angles theta=0,pi,2pi/3 about the weight-basis axis.  Set

 W=I-(H_n(g)+(3n/2)I)/(16), B=16W .

Lemma~lem:energy gives norm Wle1, norm Ble16.  In a fixed
total-weight sector the vectors are words
xin{-1,0,1}^n of fixed sum_jx_j, the diagonal of B is
[
 B_{x,x}=16-(3n)/(2)-sum_{j=0}^{n-1}x_{j-1}x_j ,
]
and for each j the move y_j=x_j-delta, y_k=x_k+delta
(k=j+1bmod n, delta=pm1, letters in {-1,0,1}) has entry -1,
except at the seam j=n-1,k=0 where it is -e^{ithetadelta} (the
phase (eq:seam-phase)).  On a coordinate u+vomega the cyclic
twist acts by

 -omega(u+vomega)=v+(v-u)omega,
 -omega^{-1}(u+vomega)=(u-v)+uomega,

so every matrix-vector product is integer arithmetic in two coordinates.

Only some columns are needed.  Let s(x)=(x_{n-1},x_0,...,x_{n-2}).
The seam formula gives H_{s(x),s(y)}=e^{itheta(y_{n-1}-x_{n-1})}H_{x,y},
so |x>|-> e^{-itheta x_{n-1}}|s(x)> commutes with H.
Reversal of the word and global negation of letters both send H to its
complex conjugate, and composing either permutation with conjugation
gives an antiunitary symmetry.  Each symmetry preserves the norm of the
corresponding column of any real exponential of H.  In every sector of
nonnegative total weight take the lexicographically least word in each
orbit under rotations and reversal, give it the orbit size as
multiplicity, and double for strictly positive total weight to include
the opposite sector.  These multiplicities sum to 3^n and reproduce
the full Frobenius norm of the exact exponential.  No equivariance of the
rounded columns is needed: each computed representative is compared with
its own exact column at its own multiplicity.

## A rational polynomial and its error

Write
[
 E=e^{84(W-I)}=e^{-(b/2)(H_n(g)+(3n/2)I)}, x_n=(3/2-a)bn,
]
so Z_n(b,g)=e^{x_n}norm{E}_F^2 (Frobenius norm).  For 0le jle256
let

 p_j=frac{84^j/j!}{sum_{l=0}^{256}84^l/l!},
 P(W)=sum_{j=0}^{256}p_jW^j .

These are mean-84 Poisson probabilities conditioned to le256; for
a mean-84 Poisson variable V,
[
 Pr(V>256)lefrac{E(2^V)}{2^{257}}=frac{e^{84}}{2^{257}}
 <frac{27^{28}}{2^{257}}<2^{-117},
]
and since norm{W^j}le1 conditioning changes the operator average by
at most twice this probability (Fox--Glynn coefficient argument
cite[Proposition~1]{FoxGlynn1988}):

 norm{P(W)-E}<2^{-116}.


Put Q=2^{56}, q_j=floor( Qp_j).  From z=0, for
j=256,...,0:

 z←fl(Bz/16)+q_je_r ,

with fl the floor on each real coordinate (on u,v
separately for u+vomega).  The ideal suffix at step j is
Qsum_{l=j}^{256}p_lW^{l-j}e_r of norm at most Q, and one
coordinate floor adds less than 2√(d_0) in a sector of dimension
d_0 while the coefficient floor adds at most one; contraction of W
bounds the error at every stage by

 257(2√(3^n)+1)le125159.

The zero coefficients above q_{172} may be skipped without changing
this count (all 257 steps are still charged).

The repeated computed-column list, divided by Q and multiplied by
e^{x_n/2}, differs from √(Z_n(b,g)) by at most

 e^{x_n/2}√(3^n)(frac{257(2√(3^n)+1)}Q+2^{-116})
 <4·10^{-7},

because √(3^n)le243, x_n/2<6, e^{x_n/2}<3^6, and
3^6·243(125159/2^{56}+2^{-116})<4·10^{-7}.

## Exact evaluation by packed integers

For efficiency the recurrence runs on batches of at most 192 columns,
storing a digit list y_0,...,y_{w-1} as
pack(y)=sum_{t=0}^{w-1}y_t2^{64t} (arbitrary precision).
Integer linear arithmetic preserves this identity before the floor even
with carries.  For either coefficient, |u|,|v|le2|u+vomega|, so by
(eq:thermal120-column-error) every coordinate of Bz is at most

 32(Q+125159)=2305843009217699040<2^{63}.

Let s_w=sum_{t=0}^{w-1}2^{64t} and
bias=2^{63}s_w, mask=(2^{60}-1)s_w,
bias16=2^{59}s_w.  For Y=pack(y) obeying
(eq:thermal120-lane-bound), the operation
[
 texttt{(((Y+bias)>{}>4)&mask)-bias16}
]
equals pack((floor( y_t/16))_t).  Adding the
bias makes every base-2^{64} digit lie in [0,2^{64}); the global
shift moves four low bits of each higher lane into the four high bits of
the previous lane; the mask deletes exactly those crossing bits, keeping
floor((y_t+2^{63})/16) in lane t; subtracting 2^{59} gives
floor( y_t/16), also for negative y_t.  Hence the packed
recurrence equals (eq:thermal120-horner).  Final coordinates obey
2(Q+125159)<2^{63}, so biasing and unpacking recovers every integer
exactly; for u+vomega the squared modulus is u^2-uv+v^2, and summing
these integers with representative multiplicities gives the integer
t_{n,g}.

## Exact totals and trace enclosures

The integer recurrence gives:

{ccl}

n&g&t_{n,g}

6&1&93637037510293158533967385845216
6&P&3629519998654831111446492683663
6&C&9839536903588146388838689926305
8&1&4947327904487991279560670127283
8&P&749999795795292705866802820621
8&C&1301346729026243689610427021457
10&1&388818111664728958208623998490
10&P&120458568770462253326886628639
10&C&167487042402286114495695779656



Enclose the scalar factor by
[
 L_n=sum_{j=0}^{100}(x_n^j)/(j!),
 U_n=L_n+2frac{x_n^{101}}{101!}.
]
Since 0<x_n<12 and later terms have ratio at most x_n/102<1/2,
L_n<e^{x_n}<U_n.  For each center c_{n,g} of the b=21/2 table,
exact rational comparison gives

 c_{n,g}-2·10^{-6}
 <frac{L_nt_{n,g}}{Q^2}
 <frac{U_nt_{n,g}}{Q^2}
 <c_{n,g}+2·10^{-6}.

These put every scaled norm below 4, so by
(eq:thermal120-scaled-error) replacing its square by the true trace
costs less than (8+4·10^{-7})4·10^{-7}.  Together with
(eq:thermal120-rational-enclosure),
[
 |Z_n(21/2,g)-c_{n,g}|
 <2·10^{-6}+(8+4·10^{-7})4·10^{-7}<10^{-5}.
]
This proves the b=21/2 table.

# Thermal calculation at inverse temperature 49/4

We prove the b=49/4 table, using integer approximations to columns of a
physical matrix exponential.  Throughout
[
 b=(49)/(4), J=280, Q=2^{55}, 4le nle12,
]
with twists the identity and rotation by pi about z (the latter has
the same partition function as P by a global rotation).

## The physical matrix and its rational approximation

In the weight basis a word w=(w_0,...,w_{n-1}), w_iin{-1,0,1},
has conserved total weight.  On each weight block set

 R=I-(frac32nI+H)/(16), D=32R ,

an integer matrix with diagonal
D_{w,w}=32-3n-2sum_{i=0}^{n-1}w_{i-1}w_i, an entry -2 for an allowed
opposite unit step at letters i-1,i, and +2 on the twisted seam
i=0 (indices mod n).  Lemma~lem:energy gives
[
 0le3/2nI+Hle5/2nI,
 spec(R)⊆[1-5n/32,1]⊆[-7/8,1],
]
so R is a self-adjoint contraction and
[
 F=exp![-tfrac b2(3/2nI+H)]=exp[98(R-I)] .
]
On the full physical space (direct sum over weight blocks),

 Z_n(b,g)=e^{x_n}norm{F}_F^2, x_n=bn(3/2-a).

Define the conditioned Poisson probabilities

 p_j=frac{v_j}{sum_{ell=0}^Jv_ell},
 v_j=98^j(J!)/(j!), 0le jle J .

Since norm Rle1, discarding the Poisson tail and renormalizing
changes the exponential by at most twice the tail probability
(Fox--Glynn):
[
 norm{F-sum_{j=0}^Jp_jR^j}
 le2e^{-98}sum_{j=281}^∞(98^j)/(j!)
 le2^{-280}e^{98}<2^{-84}<Q^{-1},
]
using 2^jge2^{281} in the tail and e^{-98}sum_{jge0}196^j/j!=e^{98},
then e<4.

## Integer columns and their error

For a unit basis column e_r, start at X=0 and for j=J,...,0 do

 X←Big⌊((DX)/(32))+c_je_r,
 c_j=floor( Qp_j),

with coordinatewise floor; all quantities are integers, and since
c_j=0 for j>191 the initial zero steps may be omitted.  Compare with
the ideal suffix Y_j=Qsum_{ell=j}^Jp_ell R^{ell-j}e_r,
norm{Y_j}_2le Q.  In a block of dimension d_0 a vector floor adds
less than √(d_0) and a coefficient floor less than one, so
contraction bounds the accumulated error by 281(√(d_0)+1).  With

 d_n=floor(√(3^n))+1,
 E_n=281(d_n+1)+1, E_n=d_nE_n,

every computed column obeys

 norm{X-QFe_r}_2le E_n .

The bound charges all 281 ideal steps (including the omitted ones).

The recursion is exactly signed-64-bit arithmetic: every coordinate is
at most Q+281(d_n+1)<2Q with d_nle730; any row of D has absolute
sum at most |32-3n|+2n+4nle76.  Hence every product and partial sum is
below 76·2Q=5476377146882523136<2^{63}-1, and division by 32 plus
c_jle Q stays below 207165582859042816<2^{63}-1.  Starting at X=0
the bounds establish exactness of the next integer operation, and
contraction establishes the next coordinate bound; the induction does not
assume the overflow-freeness it proves.  Final squares and sums use
arbitrary-precision integers.

## Representative columns

One column per orbit suffices.  For positive total weight use the orbit
under cyclic shifts and reversal and give the representative twice the
orbit size (this includes the uncomputed negative-weight block); in the
zero-weight block include digit negation in the orbit and use its size
without a factor of two.  Take the lexicographically least word,
handle periodic words by set cardinalities; the multiplicities nu_r
satisfy sum_rnu_r=3^n.

Symmetries: reversal fixes the seam as an unordered bond and global
negation preserves every entry.  For the twisted chain a bare shift moves
the seam; left rotation by q moves it to (k-1,k), k=-qbmod n, and
the diagonal unitary
[
 G_k|w>=(-1)^{sum_{j=0}^{k-1}w_j}|w>, G_0=I,
]
flips the hopping signs exactly at the two boundary bonds of that arc,
restoring the seam.  The resulting signed shift commutes with H, so
exact exponential columns at rotated words have equal norms; the
untwisted chain needs no correction.

Let X_r be the output of (eq:thermal72-horner) at representative
r and

 W_{n,g}=sum_rnu_rsum_wX_r(w)^2 .

Rounding need not preserve the signed symmetries, so form abstract direct
sums repeating each exact (resp. computed) representative column nu_r
times: the first has squared norm Q^2norm{F}_F^2, the second W_{n,g}.
By (eq:thermal72-column-error),

 |sqrt{W_{n,g}}-Qnorm{F}_F|le√(3^n)E_n<E_n,

which proves the full trace estimate without equivariance of the rounded
columns.

## Exact totals and rational enclosures

Equations (eq:thermal72-coefficients),
(eq:thermal72-horner), (eq:thermal72-total) give (second
column: no twist; third: pi twist):

begingroup
footnotesize

{r|rr}
(n)&(W_{n,I})&(W_{n,P}) 
4&1298092848946206529211955779616040&683496640248050713756440676071
5&9095379218684502088743241789&9205948219074287936685289758037
6&11971284263768914176431179720622&266573596119930013833919176645
7&15044150708248976451784593468&534330762492192297990602166370
8&386135473475360677201628357188&42459883501386986922766507327
9&4866417737994347551816846903&37716485750353424806876104310
10&19772960374215472827775892486&5030484777222468978572235564
11&841347687722481981930750335&2928767918985354021900689432
12&1266666171463993322695232916&524785463625387865619045258

endgroup


Turn these integers into the b=49/4 table.  Let
[
 r_{n,g}=floor(sqrt{W_{n,g}}),
 s_n=sum_{j=0}^{100}(x_n^j)/(j!),
 t_n=frac{x_n^{100}}{100!}.
]
Since 0le x_n<15, later omitted terms have ratio at most 15/101 and
the tail is at most 15t_n/86<t_n, so s_nle e^{x_n}le s_n+t_n.
Equations (eq:thermal72-trace) and (eq:thermal72-frobenius)
give

 L_{n,g}=frac{s_nmax(0,r_{n,g}-E_n)^2}{Q^2},
 U_{n,g}=frac{(s_n+t_n)(r_{n,g}+1+E_n)^2}{Q^2},
 L_{n,g}le Z_n(b,g)le U_{n,g},

all rational/integer with the integer square root fixed by
r^2le W<(r+1)^2.  For the integer centers C_{n,g} of the b=49/4
table, direct evaluation yields
[
 L_{n,g}>frac{C_{n,g}-30}{10^6},
 U_{n,g}<frac{C_{n,g}+30}{10^6}(4le nle12, g=I,P),
]
each strict ineuality with margin above 13/10^6.  This gives a
reproducible finite verification of all eighteen enclosures and proves the
b=49/4 table.

# Two variational inputs

The initializations need a lower bound on the untwisted partition function
at lengths 72 and 120.  One fixed triple of integer matrices defines
periodic matrix-product vectors at both lengths with energy per bond below
-a=-700741/500000.  A block decomposition reduces the computation to
powers of integer matrices of order at most 162.

## Norm and energy contractions

For Nge4 and real Dtimes D matrices A_s (s=-1,0,1) define, in
the orthonormal spin-one weight basis,

 psi_N(s_0,...,s_{N-1})=Tr(A_{s_0}... A_{s_{N-1}}),

the periodic matrix-product representation of
cite[Section~2.1, Equations~(2)--(3)]{PerezGarcia2007}.  Put

 B_{st}&=A_sA_t, T=sum_{s=-1}^1A_s ⊗  A_s,
 
 O&=sum_{s,t=-1}^1st,B_{st} ⊗  B_{st}
 +sum_{substack{s=-1,0t=0,1}}
 (B_{st} ⊗  B_{s+1,t-1}
 +B_{s+1,t-1} ⊗  B_{st}).
 

All traces are ordinary, unnormalized.


Let Nge4, Dge1 and A_{-1},A_0,A_1in M_D(R); define
psi_N,T,O by (eq:trial-vector)--(eq:trial-insertion) and,
for the periodic spin-one Hamiltonian H_N, set
V_N=Tr(T^N), U_N=Tr(T^{N-2}O).  Then
norm{psi_N}^2=V_N and
<psi_N,H_Npsi_N>=NU_N.  In particular, if V_N>0 and
U_N<-aV_N, then E_0(N)<-Na and Z_N(b)>1 for every b>0.


Proof. 
Expanding T^N and using Tr(X ⊗  Y)=Tr(X)Tr(Y), the sum of
squares of the (real) amplitudes (eq:trial-vector) is V_N; no
self-adjointness or positivity of T is assumed.  In the weight basis a
bond has diagonal st on (s,t) and off-diagonal 1 from (s,t) to
(s+1,t-1) for s=-1,0, t=0,1 (from
h=S^z ⊗  S^z+(S^+ ⊗  S^-+S^- ⊗  S^+)/2 and ladder entries
sqrt2).  Inserting this bond into the two amplitude layers gives
exactly O; summing the other indices gives U_N for one bond, and
cyclicity of the amplitude trace makes psi_N translation invariant, so
every bond including the seam contributes equally.  For V_N>0 the
variational principle gives E_0(N)le NU_N/V_N<-Na, so the shifted
ground energy is negative and its term in Tr e^{-b(H_N+NaI)} exceeds
one for every b>0.


## One integer matrix triple

Set D=26, index eight types alpha=1,...,8 by
(ell_1,...,ell_8)=(1,1,1,1,3,3,3,5), and order virtual labels
(alpha,d) first by alpha then by
d=-ell_alpha,-ell_alpha+2,...,ell_alpha.  The matrices A_s
use the diagonal parameters D_alpha of Table~tab:trial-diagonal
and the parameters P_{alphabeta} of Table~tab:trial-pairs,
defined only when ell_beta=ell_alpha+2.

[ht]
centering
{c|rrrrrrrr}
alpha&1&2&3&4&5&6&7&8 
D_alpha&-289&-84&-9&33&-22&27&71&-11

Caption: Diagonal parameters, common to both lengths.



[ht]
centering
{c|r@{}c|r@{}c|r}
(alpha,beta)&P_{alphabeta}&
(alpha,beta)&P_{alphabeta}&
(alpha,beta)&P_{alphabeta} 
(1,5)&-11&(2,7)&-49&(4,6)&9
(1,6)&-35&(3,5)&-6&(4,7)&-32
(1,7)&-28&(3,6)&-25&(5,8)&1
(2,5)&-23&(3,7)&-20&(6,8)&2
(2,6)&16&(4,5)&-34&(7,8)&-16

Caption: Parameters joining virtual types whose ell differ by 2.



Entry rules.  Row (alpha,d), column (beta,e), with ell=ell_alpha,
k=ell_beta, sin{-1,0,1}.  The entry is zero unless

 e=d+2s, |k-ell|le2, (k=ell⇒alpha=beta).

For an allowed entry set
[
 p=D_alpha,& k=ell, P_{alphabeta},& k=ell+2,
 -P_{betaalpha},& k=ell-2,
]
r=(d+ell)/2, and the entry pf where

{c|ccc}
 &s=-1&s=0&s=1 
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


lists the values of f.  These rules do not depend on N.

## Finite contractions and variational bounds

Evaluate the contractions at N=72 and N=120.  The support rule
(eq:trial-support) makes this a short block calculation: on paired
virtual labels ((alpha,d),(beta,e)) both T and O preserve d-e
(the two layers shift by the same 2s for T, and by the same total
physical weight s+t for O, also in each off-diagonal term).  The
difference blocks are exhaustive with dimensions

{c|rrrrrrrrrrr}
d-e&-10&-8&-6&-4&-2&0&2&4&6&8&10 
dim&1&8&32&80&136&162&136&80&32&8&1
,

summing to 676=26^2, each block once.

The whole integer computation: form A_s, B_{st}, T, O from the
tables and (eq:trial-transfer), (eq:trial-insertion),
(eq:trial-support), (eq:trial-entry); restrict to each
difference block giving T_delta,O_delta; compute powers by
[
 P_0=I, P_{2j}=P_j^2 (jge1), P_{2j+1}=P_j^2T_delta (jge0),
]
then accumulate

 U_N=sum_deltasum_{i,j}(P_{N-2})_{ij}(O_delta)_{ji},
 V_N=sum_deltasum_{i,j}(P_{N-2})_{ij}(T_delta^2)_{ji},

using only arbitrary-precision integers.  For N=72 the power is 70;
for N=120 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, -c_NV_Nle10^{12}U_N<(-c_N+1)V_N,
 
 {c|r}
 N&c_N 
 72&1401482112601
 120&1401482105174
 .



For the untwisted periodic spin-one Heisenberg Hamiltonian and
a=700741/500000, E_0(72)<-72a and E_0(120)<-120a; consequently
Z_{72}(b)>1 and Z_{120}(b)>1 for every b>0.


Proof. 
For each row of (eq:trial-integer-results),
(-c_N+1)/10^{12}<-700741/500000, so the corresponding explicit matrices
have V_N>0 and U_N/V_N<-a; apply Lemma~lem:trial-contraction.


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

{99}
bibitem[Haldane(1981)]{Haldane1981} F.~D.~M. Haldane, Ground state properties of antiferromagnetic chains with unrestricted spin, ILL preprint SP-81/95 (1981); reproduced as arXiv:1612.00076.
bibitem[Haldane(1983a)]{Haldane1983PLA} F.~D.~M. Haldane, Continuum dynamics of the 1-D Heisenberg antiferromagnet, Phys. Lett. A 93, 464--468 (1983).
bibitem[Haldane(1983b)]{Haldane1983PRL} F.~D.~M. Haldane, Nonlinear field theory of large-spin Heisenberg antiferromagnets, Phys. Rev. Lett. 50, 1153--1156 (1983).
bibitem[Lieb, Schultz and Mattis(1961)]{LiebSchultzMattis1961} E. Lieb, T. Schultz and D. Mattis, Two soluble models of an antiferromagnetic chain, Ann. Phys. 16, 407--466 (1961).
bibitem[Affleck and Lieb(1986)]{AffleckLieb1986} I. Affleck and E.~H. Lieb, A proof of part of Haldane's conjecture on spin chains, Lett. Math. Phys. 12, 57--69 (1986).
bibitem[Affleck, Kennedy, Lieb and Tasaki(1987)]{AKLT1987} I. Affleck, T. Kennedy, E.~H. Lieb and H. Tasaki, Rigorous results on valence-bond ground states in antiferromagnets, Phys. Rev. Lett. 59, 799--802 (1987).
bibitem[Affleck, Kennedy, Lieb and Tasaki(1988)]{AKLT1988} I. Affleck, T. Kennedy, E.~H. Lieb and H. Tasaki, Valence bond ground states in isotropic quantum antiferromagnets, Comm. Math. Phys. 115, 477--528 (1988).
bibitem[Yarotsky(2006)]{Yarotsky2006} D.~A. Yarotsky, Ground states in relatively bounded quantum perturbations of classical lattice systems, Comm. Math. Phys. 261, 799--819 (2006); arXiv:math-ph/0412040.
bibitem[White and Huse(1993)]{WhiteHuse1993} S.~R. White and D.~A. Huse, Numerical renormalization-group study of low-lying eigenstates of the antiferromagnetic S=1 Heisenberg chain, Phys. Rev. B 48, 3844--3852 (1993).
bibitem[Buyers et al.(1986)]{Buyers1986} W.~J.~L. Buyers, R.~M. Morra, R.~L. Armstrong, M.~J. Hogan, P. Gerlach and K. Hirakawa, Experimental evidence for the Haldane gap in a spin-1 nearly isotropic antiferromagnetic chain, Phys. Rev. Lett. 56, 371--374 (1986).
bibitem[Batista and Ortiz(2001)]{BatistaOrtiz2001} C.~D. Batista and G. Ortiz, Generalized Jordan--Wigner transformations, Phys. Rev. Lett. 86, 1082--1085 (2001); arXiv:cond-mat/0008374.
bibitem[Tasaki(2025)]{Tasaki2025} H. Tasaki, Ground state of the S=1 antiferromagnetic Heisenberg chain is topologically nontrivial if gapped, Phys. Rev. Lett. 134, 076602 (2025); arXiv:2407.17041 (v4).
bibitem[Fannes, Nachtergaele and Werner(1992)]{FNW1992} M. Fannes, B. Nachtergaele and R.~F. Werner, Finitely correlated states on quantum spin chains, Comm. Math. Phys. 144, 443--490 (1992).
bibitem[P'erez-Garc'ia et al.(2007)]{PerezGarcia2007} D. P'erez-Garc'ia, F. Verstraete, M.~M. Wolf and J.~I. Cirac, Matrix product state representations, Quantum Inf. Comput. 7, 401--430 (2007); arXiv:quant-ph/0608197.
bibitem[Knabe(1988)]{Knabe1988} S. Knabe, Energy gaps and elementary excitations for certain VBS-quantum antiferromagnets, J. Stat. Phys. 52, 627--638 (1988).
bibitem[Nachtergaele(1996)]{Nachtergaele1996} B. Nachtergaele, The spectral gap for some spin chains with discrete symmetry breaking, Comm. Math. Phys. 175, 565--606 (1996); arXiv:cond-mat/9410110.
bibitem[Gosset and Mozgunov(2016)]{GossetMozgunov2016} D. Gosset and E. Mozgunov, Local gap threshold for frustration-free spin systems, J. Math. Phys. 57, 091901 (2016); arXiv:1512.00088.
bibitem[Banerjee, Guo and Chowdhury(2026)]{BanerjeeGuoChowdhury2026} S. Banerjee, H. Guo and D. Chowdhury, Bootstrap certification of string order in quantum spin chains, arXiv:2609.22420 (2026).
bibitem[Suzuki(1985)]{Suzuki1985} M. Suzuki, Transfer-matrix method and Monte Carlo simulation in quantum spin systems, Phys. Rev. B 31, 2957--2965 (1985).
bibitem[Aizenman and Nachtergaele(1994)]{AizenmanNachtergaele1994} M. Aizenman and B. Nachtergaele, Geometric aspects of quantum spin states, Comm. Math. Phys. 164, 17--63 (1994); arXiv:cond-mat/9310009.
bibitem[Fox and Glynn(1988)]{FoxGlynn1988} B.~L. Fox and P.~W. Glynn, Computing Poisson probabilities, Comm. ACM 31, 440--445 (1988).
bibitem[Bochev and Markov(1989)]{BochevMarkov1989} P. Bochev and S. Markov, A self-validating numerical method for the matrix exponential, Computing 43, 59--72 (1989).
bibitem[Goldsztejn and Neumaier(2014)]{GoldsztejnNeumaier2014} A. Goldsztejn and A. Neumaier, On the exponentiation of interval matrices, Reliable Computing 20, 53--72 (2014).