diff --git a/spin_one_periodic_haldane_v2.txt b/spin_one_periodic_haldane_v2.txt
index b2fff58..aa0a662 100644
--- a/spin_one_periodic_haldane_v2.txt
+++ b/spin_one_periodic_haldane_v2.txt
@@ -26,7 +26,7 @@
 # The periodic spin-one Haldane gap
 
 
-\maketitle
+maketitle
 Abstract: 
 We prove a uniform positive spectral gap for the antiferromagnetic
 spin-one Heisenberg chain on even periodic rings, in the
@@ -46,27 +46,27 @@ for the pure bilinear model on even periodic rings, with bounds that
 are uniform in the ring length.
 
 
-Let $S^x,S^y,S^z\in M_3(\C)$ be self-adjoint spin-one matrices with
-\[
- [S^\alpha,S^\beta]=i\textstyle\sum_\etaε_{\alpha\beta\eta}S^\eta,
- \qquad \sum_\alpha(S^\alpha)^2=2I_3 .
-\]
-For even $L\ge4$ put $\cH_L=(\C^3)^{ ⊗  L}$, and on it
+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,
+ qquad 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,
- \qquad S_L^\alpha=S_0^\alpha .
+ H_L=sum_{j=0}^{L-1}sum_{alpha=x,y,z}S_j^alpha S_{j+1}^alpha,
+ qquad 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 $\Delta_1=liminf_{L\to∞,
-L even}\gamma_L$.  Then the ground state of $H_L$ is unique
-for every even $L\ge60$, and
+$gamma_L=E_1(L)-E_0(L)$, 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) && (L\ge60 even),
- \\
- \gamma_L&>(\log 20)/(784) && (L\ge2304 even),
+ gamma_L&>frac4{105}log(80)/(79) && (Lge60 even),
+ 
+ gamma_L&>(log 20)/(784) && (Lge2304 even),
  
 
-so $\Delta_1\ge\log(20)/784>0$.  The same conclusions hold for any
+so $Delta_1gelog(20)/784>0$.  The same conclusions hold for any
 simultaneously similar spin-one realization satisfying the same
 commutator and Casimir normalizations.
 
@@ -75,19 +75,19 @@ 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.
+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
+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
+(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
+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
@@ -115,7 +115,7 @@ eigenvalue estimate is a premise.
 
 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
+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
@@ -123,13 +123,13 @@ given interaction.
 
 The Affleck--Kennedy--Lieb--Tasaki construction gave a rigorous
 gapped spin-one chain with biquadratic interactions
-[cite].  Its bond interaction is $h+\tfrac13h^2$ up
-to a scalar, with $h=\mathbf S_j\cdot\mathbf S_{j+1}$, whereas
+[cite].  Its bond interaction is $h+tfrac13h^2$ up
+to a scalar, with $h=mathbf S_jbfS_jcdotmathbf S_bfS_{j+1}$, 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
+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
@@ -142,7 +142,7 @@ 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
+cite[arXiv v4, Assumption~2 and Theorem~3]{Tasaki2025}; his boundary
 hypotheses require an argument separate from the even-periodic
 estimates here.
 
@@ -151,7 +151,7 @@ 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
+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
@@ -159,11 +159,11 @@ energies of periodic matrix-product vectors
 matrix-exponential traces; validated matrix exponentiation is
 classical [cite], and the
 conditioned Poisson polynomial uses the Fox--Glynn estimate
-\cite[Proposition~1]{FoxGlynn1988}.
+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
+cite[arXiv v1, Section~2.2]{AizenmanNachtergaele1994}, followed by a
 direct operator construction on bond histories.
 
 # The model and its twisted partition functions
@@ -176,11 +176,11 @@ We use self-adjoint spin-one matrices throughout; the non-self-adjoint
 case reduces to this by the next lemma.
 
 [Representation]
-Let $S^x,S^y,S^z\in M_3(\C)$ satisfy
-\[
- [S^\alpha,S^\beta]=i\textstyle\sum_\gammaε_{\alpha\beta\gamma}S^\gamma,
- \qquad \sum_\alpha(S^\alpha)^2=2I .
-\]
+Let $S^x,S^y,S^zin M_3(C)$ satisfy
+[
+ [S^alpha,S^beta]=itextstylesum_gammaε_{alphabetagamma}S^gamma,
+ qquad 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
@@ -188,92 +188,92 @@ periodic $H_n$.
 
 
 Proof. 
-Put $S^\pm=S^x\pm iS^y$.  The assumptions give
+Put $S^pm=S^xpm iS^y$.  The assumptions give
 
- [S^z,S^\pm]=\pm S^\pm,\qquad
- S^-S^+=2I-(S^z)^2-S^z,\qquad S^+S^-=2I-(S^z)^2+S^z .
+ [S^z,S^pm]=pm S^pm,qquad
+ S^-S^+=2I-(S^z)^2-S^z,qquad 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
+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.
+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 $|s>$, $s\in\{-1,0,1\}$, the generator is
-$S^z|s>=s|s>$ with ladder entries $\sqrt2$.  In the
+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
-\[
- (S^\alpha)_{ij}=-iε_{\alpha i j},\qquad G_\alpha=iS^\alpha
-\]
-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
+[
+ (S^alpha)_{ij}=-iε_{alpha i j},qquad 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,\qquad
- h=F-|Ω><Ω|,\quad
- Ω=\sum_{i=1}^3|ii>,
+ -h=sum_alpha G_alpha ⊗  G_alpha,qquad
+ h=F-|Ω><Ω|,quad
+ Ω=sum_{i=1}^3|ii>,
 
 where $F(u ⊗  v)=v ⊗  u$.  Hence $h$ has eigenvalues
-$-2,-1,1$ and $h\le I$.
+$-2,-1,1$ and $hle I$.
 
 ## Twists and partition functions
 
-Number the sites $0,...,n-1$ with $n\ge4$.  A proper rotation
-$g\in SO(3)$ is unitary on $\C^3$.  Define
+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),\qquad
- h_{n-1,0}(g)=\sum_\alpha S_{n-1}^\alpha\,(gS^\alpha g^{-1})_0 ,
+ 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 ,
 
 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),\qquad
- Z_n(b,g)=\Tr\exp[-b(H_n(g)+anI)],\qquad Z_n(b)=Z_n(b,I),
+ a=(700741)/(500000),qquad
+ Z_n(b,g)=Trexp[-b(H_n(g)+anI)],qquad 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
-\[
- P_x=diag(1,-1,-1),\quad
- P_y=diag(-1,1,-1),\quad
+[
+ P_x=diag(1,-1,-1),quad
+ P_y=diag(-1,1,-1),quad
  P_z=diag(-1,-1,1),
-\]
-$D_2=\{I,P_x,P_y,P_z\}$, and the cyclic rotation
-$C:e_x\mapsto e_y\mapsto e_z\mapsto e_x$; write $P=P_x$.  By a global
+]
+$D_2={I,P_x,P_y,P_z}$, and the cyclic rotation
+$C:e_xmapsto e_ymapsto e_zmapsto e_x$; 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 $\exp(-i\theta S^z)$ with
-$\theta=\pi$ and $\theta=2\pi/3$.
+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
+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\!(i\theta(y_0-x_0)),
+ exp!(itheta(y_0-x_0)),
 
-since $gS^\pm g^{-1}=e^{\mp i\theta}S^\pm$.  These formulas tie the
+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 $n\ge4$ and $g\in SO(3)$, $-(3n)/(2)I\le H_n(g)\le nI$.
+For every $nge4$ and $gin SO(3)$, $-(3n)/(2)Ile H_n(g)le nI$.
 
 
 Proof. 
-The upper bound is $h\le I$ plus unitary conjugation of the seam.  For
+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 unequal pair
 swaps with coefficient $-1$, an equal pair changes to either other equal
@@ -282,24 +282,24 @@ three-letter word a positive weight $q$ according to its equality
 pattern:
 
 {c|c|c}
-word type & $q$ & absolute weighted row sum\\ \hline
-$aaa$ & $4$ & $3+3+3+3=12$\\
-$aab$ or $abb$ & $3$ & $4+3+2=9$\\
-$aba$ & $2$ & $3+3=6$\\
+word type & $q$ & absolute weighted row sum hline
+$aaa$ & $4$ & $3+3+3+3=12$
+$aab$ 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,
-$h_{12}+h_{23}\ge-3I$.
+$h_{12}+h_{23}ge-3I$.
 
 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 inequalities counts each bond twice and gives
-$H_n(g)\ge-(3n)/(2)I$.
+$H_n(g)ge-(3n)/(2)I$.
 
 
 The trial inputs of Appendix~sec:trials give $E_0(n)+an<0$ at
@@ -313,89 +313,89 @@ the physical partition function (eq:shift).  This is what turns
 even spatial powers into positive weights.  The operator itself need
 not be positive.
 
-Let $\mathcal G\subset SO(3)$ be the group of proper signed permutation
-matrices (it contains $D_2$ and $C$).  For $g\in\mathcal G$ write
-\[
- g e_\alpha=ε_g(\alpha)e_{\sigma_g(\alpha)},
- \qquad ε_g(\alpha)\in\{-1,1\}.
-\]
+Let $mathcal GcalGsubset SO(3)$ be the group of proper signed permutation
+matrices (it contains $D_2$ and $C$).  For $ginmathcal G$calG$ write
+[
+ g e_alpha=ε_g(alpha)e_{sigma_g(alpha)},
+ qquad ε_g(alpha)in{-1,1}.
+]
 Rotation covariance of the Cartesian matrices gives
 
- gG_\alpha g^{-1}=ε_g(\alpha)G_{\sigma_g(\alpha)} .
+ gG_alpha g^{-1}=ε_g(alpha)G_{sigma_g(alpha)} .
 
 
 [Spatial transfer]
-For each $b>0$ there are a Hilbert space $\mathcal K_b$, a compact
+For each $b>0$ there are a Hilbert space $mathcal K_b$,calK_b$, a compact
 self-adjoint operator $X_b$, and a unitary representation
-$g\mapsto U_g$ of $\mathcal G$ commuting with $X_b$, such that
+$gmapsto U_g$ of $mathcal G$calG$ commuting with $X_b$, such that
 
- Z_n(b,g)=\Tr(X_b^nU_g)\qquad(n\ge4,\ g\in\mathcal G).
+ Z_n(b,g)=Tr(X_b^nU_g)qquad(nge4, ginmathcal G).calG).
 
 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
-$k\ge2$.  If $\lambda_i(b)$ are the nonzero eigenvalues, counted with
+$kge2$.  If $lambda_i(b)$ are the nonzero eigenvalues, counted with
 multiplicity, then
 
- \sum_i|\lambda_i(b)|^k<∞\ (k\ge2),\qquad
- Z_n(b)=\sum_i\lambda_i(b)^n\ (n\ge4).
+ sum_i|lambda_i(b)|^k<∞ (kge2),qquad
+ Z_n(b)=sum_ilambda_i(b)^n (nge4).
 
 
 
 Proof. 
 Histories and kernel.  A bond history is a finite ordered list
-\[
- x=((t_1,\alpha_1),...,(t_k,\alpha_k)),\quad
- 0<t_1<...<t_k<b,\ \alpha_j\in\{x,y,z\},
-\]
+[
+ x=((t_1,alpha_1),...,(t_k,alpha_k)),quad
+ 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_{k\ge0}(3^kb^k)/(k!)=e^{3b},
- \qquad \mathcal K_b=L^2(Ω_b,\mu;\C).
+ mu(Ω_b)=sum_{kge0}(3^kb^k)/(k!)=e^{3b},
+ qquad mathcal K_b=L^2(Ω_b,calK_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
+$G_alpha$ (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
-$\norm{K_b}_{HS}^2\le9e^{6b}$.  Set $X_b=e^{-ab}K_b$; compact
+$norm{K_b}_{HS}^2le9e^{6b}$.  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 $k\ge2$.
+and boundedness gives $sum_i|lambda_i|^k<∞$ for $kge2$.
 
-Rotations.  The permutation $\sigma_g$ acts on labels preserving
-times and measure.  Put $s_g(x)=\prod_{\alpha\in x}ε_g(\alpha)$
-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
+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
+$k_b(sigma_gx,sigma_gy)=s_g(x)s_g(y)k_b(x,y)$, 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).
+ (K_bU_g)(x,y)=k_b(x,sigma_gy)s_g(y).
 
 
-The cycle trace.  For $n\ge2$ both $K_b^{n-1}$ and $K_bU_g$ are
+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 $\Tr(AB)=\int a(x,y)b(y,x)\,d\mu x\,d\mu y$ and
+$k_b$ 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)\,d\mu^n ,
+ 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
+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^n\sum_{k\ge0}((3nb)^k)/(k!)=3^ne^{3nb}.
+ 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
@@ -403,22 +403,22 @@ 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
-\[
- s_g(e_0)k_b(\sigma_ge_0,e_1)k_b(e_1,e_2)... k_b(e_{n-1},e_0),
-\]
+[
+ 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
+$Tr e^{-bH_n(g)}=Tr(K_b^nU_g)$, and multiplying by $e^{-abn}$ gives
 (eq:spatial-trace); the compact trace formula gives
 (eq:spatial-moments).
 
 
-For even $n\ge4$, (eq:spatial-trace) supplies the probabilities
+For even $nge4$, (eq:spatial-trace) supplies the probabilities
 
- p_i=(|\lambda_i(b)|^n)/(Z_n(b)),\qquad
- \sum_i p_i=1,\qquad
- \sum_i p_i^2=\frac{Z_{2n}(b)}{Z_n(b)^2}.
+ p_i=(|lambda_i(b)|^n)/(Z_n(b)),qquad
+ sum_i p_i=1,qquad
+ 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
@@ -432,11 +432,11 @@ 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 $k\ge2$, with
+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\qquad(n\ge4).
+ 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_nqquad(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
@@ -446,12 +446,12 @@ 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
-$\Pi_\chi=\frac14\sum_{g\in D_2}\chi(g)U_g$ commuting with $X_b$ and
+$Pi_chi=frac14sum_{gin D_2}chi(g)U_g$ 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 $1,\omega,\bar\omega$
-of $U_C$ ($\omega=e^{2\pi i/3}$); these reduce $X_b$, complex
+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
@@ -461,8 +461,8 @@ 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
-$1,\omega,\bar\omega$ on the remaining three with
-$\omega+\bar\omega=-1$, giving (eq:sector-cyclic).  The $J_n$
+$1,omega,baromega$ 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.
 
@@ -483,9 +483,9 @@ 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 $n\ge4$,
+even $nge4$,
 
- S(n,b)=\frac{Z_{2n}(b)}{Z_n(b)^2},\qquad
+ S(n,b)=frac{Z_{2n}(b)}{Z_n(b)^2},qquad
  T(n,b)=(Z_n(2b))/(Z_n(b)^2).
 
 The shift $aL$ cancels from both ratios.  First, the inequality behind
@@ -493,34 +493,34 @@ 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 $P=\sum_iw_i^2\in(0,1]$.  The normalized
+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),\qquad f(u)=(u^2)/(2(1-u)^2)\ \ (0\le u<1),
+ 1-P'le f(1-P),qquad f(u)=(u^2)/(2(1-u)^2)  (0le u<1),
 
 and $f$ is increasing.  Consequently
 
- 0<S(n,b),T(n,b)\le1,\quad
- 1-S(2n,b)\le f(1-S(n,b)),\quad
- 1-T(n,2b)\le f(1-T(n,b)).
+ 0<S(n,b),T(n,b)le1,quad
+ 1-S(2n,b)le f(1-S(n,b)),quad
+ 1-T(n,2b)le f(1-T(n,b)).
 
 
 
 Proof. 
-Some weight is positive, so $P>0$, and $w_i^2\le w_i$ gives $P\le1$.
+Some weight is positive, so $P>0$, and $w_i^2le w_i$ gives $Ple1$.
 Nonnegativity allows rearrangement of all countable sums, so
-\[
- 1-P'=\frac{2\sum_{i<j}w_i^2w_j^2}{P^2}
- \le\frac{2(\sum_{i<j}w_iw_j)^2}{P^2}
+[
+ 1-P'=frac{2sum_{i<j}w_i^2w_j^2}{P^2}
+ lefrac{2(sum_{i<j}w_iw_j)^2}{P^2}
  =((1-P)^2)/(2P^2),
-\]
-and $f'(u)=u/(1-u)^3\ge0$.  For $S(n,b)$ use the spatial
-probabilities $|\lambda_i(b)|^n/Z_n(b)$ of
+]
+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)} .
+ 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 $2b$.
 
@@ -531,7 +531,7 @@ 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).
    
 
@@ -539,20 +539,20 @@ Both denominators are at most one, and the following form lets the two
 initial bounds differ and be rounded upward.
 
 [Coupled update]
-Suppose $n\ge4$ even, $b>0$, and $p,q\in[0,1)$ with
-$1-S(n,b)\le p$, $1-T(2n,b)\le q$.  If $p_+,q_+\in[0,1)$ obey
+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)),\qquad
- q_+\ge f(1-(1-q)^2(1-p_+)),
+ p_+ge f(1-(1-p)^2(1-q)),qquad
+ q_+ge f(1-(1-q)^2(1-p_+)),
 
-then $1-S(2n,2b)\le p_+$ and $1-T(4n,2b)\le q_+$.
+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
+$S(n,2b)ge(1-p)^2(1-q)$; 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
+(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.
 
@@ -566,7 +566,7 @@ decrease.
 
 Once both defects are small one common bound suffices.  Set
 
- G(u)=\frac12((1-u)^{-1}+(1-u)^{-2}+(1-u)^{-3})^2,
+ G(u)=frac12((1-u)^{-1}+(1-u)^{-2}+(1-u)^{-3})^2,
 
 increasing on $[0,1)$, with
 
@@ -575,90 +575,90 @@ increasing on $[0,1)$, with
 
 [Uniform gap from two initial purities]
 
-For every even $L\ge4$ let $H_L$ be a non-scalar finite-dimensional
-self-adjoint Hamiltonian, and suppose that $a\in\R$ and, for each
-$b>0$, a real finite or countable list $(\lambda_i(b))$ satisfy
-\[
- \sum_i|\lambda_i(b)|^4<∞,\qquad
- Z_L(b):=\Tr e^{-b(H_L+aLI)}=\sum_i|\lambda_i(b)|^L
- \quad(L\ge4 even).
-\]
+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<∞,qquad
+ Z_L(b):=Tr e^{-b(H_L+aLI)}=sum_i|lambda_i(b)|^L
+ quad(Lge4 even).
+]
 Define $S,T$ by (eq:purities) and $G$ by
-(eq:bootstrap-coefficient).  Let $n_0\ge4$ even, $b_0>0$, and let
+(eq:bootstrap-coefficient).  Let $n_0ge4$ even, $b_0>0$, and let
 $u_0,C$ satisfy
 
- 0<u_0<\tfrac16,\quad G(u_0)\le C,\quad r:=Cu_0<1,\quad C(1-3u_0)>3,
+ 0<u_0<tfrac16,quad G(u_0)le C,quad r:=Cu_0<1,quad C(1-3u_0)>3,
 
 with seeds
 
- S(n_0,b_0)\ge1-u_0,\qquad T(2n_0,b_0)\ge1-u_0 .
+ S(n_0,b_0)ge1-u_0,qquad T(2n_0,b_0)ge1-u_0 .
 
-For $j\ge0$ set
+For $jge0$ set
 
- n_j=2^jn_0,\quad b_j=2^jb_0,\quad u_j=C^{-1}r^{2^j}.
+ n_j=2^jn_0,quad b_j=2^jb_0,quad u_j=C^{-1}r^{2^j}.
 
 Then
 
- S(n_j,b_j)\ge1-u_j,\qquad T(2n_j,b_j)\ge1-u_j,
+ S(n_j,b_j)ge1-u_j,qquad T(2n_j,b_j)ge1-u_j,
 
-and for every even $L\ge n_0$ the physical ground state is unique with
+and for every even $Lge n_0$ the physical ground state is unique with
 gap
 
- \gamma_L>(\log(1/r))/(b_0).
+ gamma_L>(log(1/r))/(b_0).
 
 
 
 Proof. 
-Doubling.  By definition $u_{j+1}=Cu_j^2$, $0<u_{j+1}<u_j\le
+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
-\[
- f(1-(1-u)^3)=G(u)u^2\le Cu^2=u_{j+1}\le u .
-\]
+[
+ 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 $p$th-power sum and $q\ge p>0$, normalization gives
-$\sum a_i^q\le(\sum a_i^p)^{q/p}$ (the normalized $p$th powers are
+finite positive $p$th-power sum and $qge p>0$, normalization gives
+$sum a_i^qle(sum a_i^p)^{q/p}$ (the normalized $p$th powers are
 probabilities, whose $q/p$ powers sum to at most one).  Applied at the
-two temperatures, whenever $n\le L\le2n$ with both even,
-\[
- Z_L(b)\le Z_n(b)^{L/n},\qquad
- Z_L(2b)\ge Z_{2n}(2b)^{L/(2n)},
-\]
+two temperatures, whenever $nle Lle2n$ with both even,
+[
+ Z_L(b)le Z_n(b)^{L/n},qquad
+ 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)} .
+ 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)^3\ge1-3u_j>\tfrac12
- \qquad(n_j\le L\le2n_j,\ L even).
+ T(L,b_j)ge(1-u_j)^3ge1-3u_j>tfrac12
+ qquad(n_jle Lle2n_j, L even).
 
 
 The physical spectrum.  The largest weight in
 (eq:gibbs-weights) is at least its purity, since
-$\sum_kw_k^2\le(\max_kw_k)\sum_kw_k$, hence $>1/2$.  These weights count
+$sum_kw_k^2le(max_kw_k)sum_kw_k$, 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_j\gamma_L}
+ 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}
+ 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
-$n_j\le L\le2n_j$; the dyadic intervals cover every even $L\ge n_0$.
+$n_jle Lle2n_j$; the dyadic intervals cover every even $Lge n_0$.
 
 
 ## The two parameter choices
@@ -666,20 +666,20 @@ $n_j\le L\le2n_j$; the dyadic intervals cover every even $L\ge n_0$.
 We will prove the seeds (eq:bootstrap-seeds) for
 
 {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\\
+ 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
 
 
 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, $1-(1-u)^3\le3u$ gives for
-$0<u\le1/100$
-\[
- G(u)\le\frac9{2(1-3u)^2}\le(45000)/(9409)<5,\qquad
- \frac3{C(1-3u_0)}=(60)/(97)<1 .
-\]
-So the two rows give the bounds $(4/105)\log(80/79)$ and $(\log20)/784$
+$3/[C(1-3u_0)]=48/79<1$.  For the second, $1-(1-u)^3le3u$ gives for
+$0<ule1/100$
+[
+ G(u)lefrac9{2(1-3u)^2}le(45000)/(9409)<5,qquad
+ 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
@@ -701,7 +701,7 @@ moment also fixed by $C$, and $N_n$ for one nontrivial $D_2$ sector
 $N_n$ three times,
 
 
- J_n&=\frac{Z_n+3Z_n(P)}4,& N_n&=\frac{Z_n-Z_n(P)}4,\\
+ 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),
 
@@ -711,30 +711,30 @@ 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 $30\cdot10^{-6}$ at $b=49/4$:
-\[
+$b=21/2$ and below $30cdot10^{-6}$ at $b=49/4$:
+[
 {c|rrr}
- &\multicolumn{3}{c}{b=21/2}\\
- n&g=1&g=P&g=C\\ \hline
- 6&8945303&346734&939987\\
- 8&3741281&567167&984108\\
+ &multicolumn{3}{c}{b=21/2}
+ n&g=1&g=P&g=C hline
+ 6&8945303&346734&939987
+ 8&3741281&567167&984108
  10&2327547&721090&1002613
 
-\qquad
+qquad
 {c|rr}
- &\multicolumn{2}{c}{b=49/4}\\
- n&g=1&g=P\\ \hline
- 4&124885379&65757\\
- 5&2925&2960742\\
- 6&12870614&286599\\
- 7&54069&1920414\\
- 8&4639283&510140\\
- 9&195455&1514846\\
- 10&2654823&675419\\
- 11&377629&1314544\\
+ &multicolumn{2}{c}{b=49/4}
+ n&g=1&g=P hline
+ 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
@@ -742,7 +742,7 @@ 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,\qquad E_0(120)<-120a,
+ E_0(72)<-72a,qquad E_0(120)<-120a,
 
 so $Z_{72}(b)>1$ and $Z_{120}(b)>1$ for every $b>0$ (one Boltzmann
 factor already exceeds one).
@@ -750,41 +750,41 @@ factor already exceeds one).
 Two elementary lemmas convert the tables into spectral bounds.
 
 [Capped masses]
-Let $(t_i)$ be finite or countable with $0\le t_i\le h$ and
-$\sum_it_i\le m$, where $h>0, m\ge0$.  For $r\ge1$ put
+Let $(t_i)$ be finite or countable with $0le t_ile h$ and
+$sum_it_ile m$, where $h>0, mge0$.  For $rge1$ put
 
- \mathcal C(m,h,r)=qh^r+(m-qh)^r,\qquad q=\lfloor m/h\rfloor.
+ mathcal C(m,h,r)=qh^r+(m-qh)^r,calC(m,h,r)=qh^r+(m-qh)^r,qquad q=lfloor m/hrfloor.
 
-Then $\sum_it_i^r\le\mathcal C(m,h,r)$; without the cap,
-$\sum_it_i^r\le m^r$.
+Then $sum_it_i^rlemathcal C(m,h,r)$;calC(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 $x\mapsto x^r$ shows the sum of $r$th powers does not
+convexity of $xmapsto x^r$ shows the sum of $r$th 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 $t_i^r\le t_i(\sum_jt_j)^{r-1}$.
+case.  The uncapped bound follows from $t_i^rle t_i(sum_jt_j)^{r-1}$.
 
 
 [Polynomial filter]
-Let $ε\ge0$, $v>0$, and let $(\lambda_i)$ be a real eigenvalue
-list with moments $M_j=\sum_i\lambda_i^j$ converging absolutely, and
-$|M_j-c_j|\leε$ for the exponents occurring in a polynomial
-$F(x)=\sum_jf_jx^j\ge0$ on $\R$.  If
-\[
- B=\sum_jf_jc_j+ε\sum_j|f_j|
- \quadand\quad F(x)>B\ for |x|\ge v,
-\]
-then every $|\lambda_i|<v$.
+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.
+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
@@ -801,114 +801,114 @@ 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
-\[
- m_I=(1042655)/(10^6),\quad m_O=(40041)/(10^6),\quad
+[
+ m_I=(1042655)/(10^6),quad m_O=(40041)/(10^6),quad
  m_N=(401625)/(10^6),
-\]
-with margins $19/(12\cdot10^6)$, $7/(12\cdot10^6)$,
-$3/(4\cdot10^6)$ over centers plus $10^{-5}$.  Apply
+]
+with margins $19/(12cdot10^6)$, $7/(12cdot10^6)$,
+$3/(4cdot10^6)$ over centers plus $10^{-5}$.  Apply
 Lemma~lem:polynomial-filter to
-\[
- F_I(x)=x^6(-2+10x^2)^2,\qquad F_N(x)=x^6(-25+100x^2)^2 .
-\]
+[
+ F_I(x)=x^6(-2+10x^2)^2,qquad 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
-\[
- |\lambda|<U:=(100245)/(100000)\ (I sector),\qquad
- |\lambda|<V:=(88643)/(100000)\ (N sector),
-\]
+[
+ |lambda|<U:=(100245)/(100000) (I sector),qquad
+ |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 inequalities $U^{10}<m_I<2U^{10}$ and $V^{10}<m_N<2V^{10}$ allow the
 two-piece (eq:capped-function).  For $k=30,120$ set
-\[
- A_k=U^k+(m_I-U^{10})^{k/10},\quad B_k=m_O^{k/10},\quad
+[
+ A_k=U^k+(m_I-U^{10})^{k/10},quad B_k=m_O^{k/10},quad
  C_k=V^k+(m_N-V^{10})^{k/10},
-\]
+]
 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,\qquad
- Z_k(b_0,P),Z_k(b_0,C)\le A_k+2B_k .
+ Z_k(b_0)le A_k+2B_k+3C_k,qquad
+ 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
-\[
- D=(1450)/(1000),\quad E=(1202)/(1000),\quad
- \tau=(1225)/(1000),\quad\tau_0=(1264)/(1000),\quad
+[
+ D=(1450)/(1000),quad E=(1202)/(1000),quad
+ tau=(1225)/(1000),quadtau_0=(1264)/(1000),quad
  R_q=(684)/(10000).
-\]
+]
 The scalar estimates are
 
- m^5<D^2,\quad p^5<E^2,\quad q\ge1,\quad (q-1)^5<R_q^2,\quad
- (D+11E)/(12)\le\tau,\quad (D+3E)/(4)\le\tau_0,
+ m^5<D^2,quad p^5<E^2,quad qge1,quad (q-1)^5<R_q^2,quad
+ (D+11E)/(12)letau,quad (D+3E)/(4)letau_0,
 
 with positive margins exceeding $0.001068$, $0.000433$,
 $0.0000472$ in the first, second and fourth strict inequalities; 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
+factors $z_i$, $sum_i z_i^{5/2}le(sum_iz_i)^{5/2}$, so
 (eq:short-moment-upper)--(eq:seed60-scalars) give
-\[
- Z_{30}(b)\le D,\qquad Z_{30}(b,P),Z_{30}(b,C)\le E ;
-\]
+[
+ Z_{30}(b)le D,qquad 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 $\tau_0$.
+spatial moment are at most $tau$ and $tau_0$.
 
-Let $w_i=|\lambda_i(X_b)|^{30}$, so $\sum_iw_i\le D$, while
-(eq:trial-thermal-consequence) gives $\sum_iw_i^4=Z_{120}(b)>1$.
+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 $0\le t\le0.998$ and $w_0\ge t$.  Let $x$ be
+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
-\[
- 0\le x\le\tau-t,\quad 0\le y,\quad
- x+y\le\tau_0-t,\quad z\le D-t-x-y .
-\]
-The divisors use identical repeated lists.  Since $D>\tau_0$ the last
+[
+ 0le xletau-t,quad 0le y,quad
+ x+yletau_0-t,quad 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)\in\mathcal V_t}
- (x^2+(y^2)/(2)+((D-t-x-y)^2)/(3)),\quad
- \mathcal V_t=\{(0,0),(\tau-t,0),(\tau-t,\tau_0-\tau),(0,\tau_0-t)\}.
+ R(t)=max_{(x,y)inmathcal V_t}calV_t
+ (x^2+(y^2)/(2)+((D-t-x-y)^2)/(3)),quad
+ mathcal V_t=calV_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\\ \hline
+ t&0.88&0.99&0.998 hline
  R(t)&0.1359&0.0721&0.068404
 
-\]
+]
 and
-\[
- 1-0.99^4-R(0.88)^2=0.02093518,\qquad
+[
+ 1-0.99^4-R(0.88)^2=0.02093518,qquad
  1-0.998^4-R(0.99)^2=0.002777621984.
-\]
-If $w_0\le0.99$ the first would contradict $\sum_iw_i^4>1$, and the
-second then excludes $w_0\le0.998$.  Keeping the dominant contribution
+]
+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,
-\[
- S(60,b)=(\sum_iw_i^4)/((\sum_iw_i^2)^2)
- \ge(1+(R(0.998))/(0.998^2))^{-2}>0.8756>\frac78 .
-\]
+[
+ 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
-\[
- T(120,b)\ge(1+R_q)^{-2}>0.8760>\frac78 .
-\]
+[
+ 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 $S$.
 
@@ -917,10 +917,10 @@ spatial weights used for $S$.
 
 
 At inverse temperature $b=784$,
-\[
- 1-S(2304,b)\le(84513)/(10^7)<\frac1{100},\qquad
- 1-T(4608,b)\le(27493)/(5\cdot10^6)<\frac1{100}.
-\]
+[
+ 1-S(2304,b)le(84513)/(10^7)<frac1{100},qquad
+ 1-T(4608,b)le(27493)/(5cdot10^6)<frac1{100}.
+]
 
 
 Proof. 
@@ -928,82 +928,82 @@ 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 $ε=30\cdot10^{-6}$.
+via (eq:sector-inversion) with error at most $ε=30cdot10^{-6}$.
 Use
-\[
- F_J(x)=x^4(7+17x-280x^2-185x^3+1000x^4)^2,\quad
+[
+ F_J(x)=x^4(7+17x-280x^2-185x^3+1000x^4)^2,quad
  F_N(x)=x^4(12-394x^2+1000x^4)^2,
-\]
+]
 of degree $12$, so the table supplies every needed moment.  The summed
 upper bounds of Lemma~lem:polynomial-filter are
-\[
- B_J=318604.54848175,\qquad B_N=48169.880699 .
-\]
+[
+ B_J=318604.54848175,qquad B_N=48169.880699 .
+]
 Set $v_J=100139/100000$, $v_N=872/1000$.  Direct comparison gives
-\[
- F_J(v_J)-B_J>180,\quad F_J(-v_J)-B_J>496888,\quad
- F_N(\pm v_N)-B_N>654 .
-\]
+[
+ F_J(v_J)-B_J>180,quad F_J(-v_J)-B_J>496888,quad
+ 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 $y\ge0$,
-\[
+for $yge0$,
+[
  
- Q(1+y)&=559+2902y+5165y^2+3815y^3+1000y^4,\\
+ 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 $r\ge v_N$ (value at $v_N$ positive, derivative
+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 $k=36,72$,
+Put $ell=999/1000$ and, for $k=36,72$,
 
- B(k)=3\mathcal C(m_N,v_N^{12},k/12),\quad
- R_A(k)=B(k)+(m_J-\ell^{12})^{k/12}
+ B(k)=3mathcal C(m_N,v_N^calC(m_N,v_N^{12},k/12),quad
+ 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
-$Z_{72}(b_0)\le\mathcal C(m_J,\ell^{12},6)+B(72)<1$, with strict margin
+(with $m_J>ell^{12}$).  If all $J$ eigenvalues had modulus at most
+$ell$, Lemma~lem:cappedmass would give
+$Z_{72}(b_0)lemathcal C(m_J,calC(m_J,ell^{12},6)+B(72)<1$, with strict margin
 above $0.0693$; the trial inequality 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
+$lambda_0$ 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 $k$th 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}.
+ 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 $p_0<0.047916$,
+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
-$\lceil x\rceil_7=10^{-7}\lceil10^7x\rceil$:
-\[
- p_{j+1}=\bigl\lceil f(1-(1-p_j)^2(1-q_j))\bigr\rceil_7,
- \quad
- q_{j+1}=\bigl\lceil f(1-(1-q_j)^2(1-p_{j+1}))\bigr\rceil_7 .
-\]
+$lceil xrceil_7=10^{-7}lceil10^7xrceil$:
+[
+ p_{j+1}=bigllceil f(1-(1-p_j)^2(1-q_j))bigrrceil_7,
+ quad
+ 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\\ \hline
- 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\\
+ j&p_j&q_j hline
+ 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
-$2^6$: $36\cdot64=2304$ and $(49/4)\cdot64=784$.  The last row proves
+$2^6$: $36cdot64=2304$ and $(49/4)cdot64=784$.  The last row proves
 the proposition.
 
 
@@ -1013,18 +1013,18 @@ the proposition.
 
 Let $X_b$ be the transfer operator of Proposition~prop:transfer
 with $a=700741/500000$, and put
-\[
+[
  
- b_0=49/4,\quad b_1=49/2,\quad \ell=999/1000,\\
- U=100139/100000,\quad V=872/1000,\quad m_J=1.0657205,\quad m_N=.2783155 .
+ b_0=49/4,quad b_1=49/2,quad ell=999/1000,
+ U=100139/100000,quad V=872/1000,quad m_J=1.0657205,quad 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
+$m_N$; 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
 $b>0$.
 
@@ -1054,108 +1054,108 @@ $S(60,105/4)>7/8$ 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 inequalities 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
-$L\ge60$, proving (eq:gap60).
+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 $Cu_0=1/20$,
-\[
- \gamma_L>(\log20)/(784)\qquad(L\ge2304 even),
-\]
+[
+ gamma_L>(log20)/(784)qquad(Lge2304 even),
+]
 proving (eq:gap2304).  The limit inferior over even lengths gives
-$\Delta_1\ge\log(20)/784>0$.  Finally, Lemma~lem:representation
+$Delta_1gelog(20)/784>0$.  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 $L\ge2304$, and the finitely many even $L$ with
-$4\le L\le2302$ are finite-dimensional self-adjoint Hamiltonians, each
+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
-$\inf_{L\ge4,\,L even}\gamma_L>0$, though it gives neither a
+$inf_{Lge4,,L even}gamma_L>0$, 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
-$[H,A]=\sum_{j\in\Z}[\mathbf S_j\cdot\mathbf S_{j+1},A]$, only finitely
+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
+$[H,A]=sum_{jinZ}[mathbf S_jbfS_jcdotmathbf S_bfS_{j+1},A]$, 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
-$L\to∞$, converging on every local operator to a state $\omega$;
+$Lto∞$, 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)
+ 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
+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 $L\ge2304$, Theorem~thm:main gives a
-unique ground state $\psi_L$ and gap above $\delta=\log(20)/784$.
-Writing $\omega_L(A)=<\psi_L,A\psi_L>$, the spectral
-inequality on the orthogonal complement of $\psi_L$ gives
-\[
- \omega_L(A^*[H_L,A])\ge\delta(\omega_L(A^*A)-|\omega_L(A)|^2).
-\]
+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
+inequality 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
+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
+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
+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 $\omega=e^{2\pi i/3}$.
+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 $\theta=0,\pi,2\pi/3$ about the weight-basis axis.  Set
+angles $theta=0,pi,2pi/3$ about the weight-basis axis.  Set
 
- W=I-(H_n(g)+(3n/2)I)/(16),\qquad B=16W .
+ W=I-(H_n(g)+(3n/2)I)/(16),qquad B=16W .
 
-Lemma~lem:energy gives $\norm W\le1$, $\norm B\le16$.  In a fixed
+Lemma~lem:energy gives $norm Wle1$, $norm Ble16$.  In a fixed
 total-weight sector the vectors are words
-$x\in\{-1,0,1\}^n$ 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 $y_j=x_j-\delta$, $y_k=x_k+\delta$
-($k=j+1\bmod n$, $\delta=\pm1$, letters in $\{-1,0,1\}$) has entry $-1$,
-except at the seam $j=n-1,k=0$ where it is $-e^{i\theta\delta}$ (the
-phase (eq:seam-phase)).  On a coordinate $u+v\omega$ the cyclic
+$xin{-1,0,1}^n$ 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+v\omega)=v+(v-u)\omega,\qquad
- -\omega^{-1}(u+v\omega)=(u-v)+u\omega,
+ -omega(u+vomega)=v+(v-u)omega,qquad
+ -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 $H_{s(x),s(y)}=e^{i\theta(y_{n-1}-x_{n-1})}H_{x,y}$,
-so $|x>\mapsto e^{-i\theta x_{n-1}}|s(x)>$ commutes with $H$.
+The seam formula gives $H_{s(x),s(y)}=e^{itheta(y_{n-1}-x_{n-1})}H_{x,y}$,
+so $|x>mapsto e^{-itheta x_{n-1}}|s(x)>$ 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
@@ -1171,41 +1171,41 @@ its own exact column at its own multiplicity.
 ## A rational polynomial and its error
 
 Write
-\[
- E=e^{84(W-I)}=e^{-(b/2)(H_n(g)+(3n/2)I)},\qquad x_n=(3/2-a)bn,
-\]
-so $Z_n(b,g)=e^{x_n}\norm{E}_F^2$ (Frobenius norm).  For $0\le j\le256$
+[
+ E=e^{84(W-I)}=e^{-(b/2)(H_n(g)+(3n/2)I)},qquad 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!},\qquad
- P(W)=\sum_{j=0}^{256}p_jW^j .
+ p_j=frac{84^j/j!}{sum_{l=0}^{256}84^l/l!},qquad
+ P(W)=sum_{j=0}^{256}p_jW^j .
 
-These are mean-$84$ Poisson probabilities conditioned to $\le256$; for
+These are mean-$84$ Poisson probabilities conditioned to $le256$; for
 a mean-$84$ Poisson variable $V$,
-\[
- \Pr(V>256)\le\frac{\mathbb E(2^V)}{2^{257}}=\frac{e^{84}}{2^{257}}
- <\frac{27^{28}}{2^{257}}<2^{-117},
-\]
-and since $\norm{W^j}\le1$ conditioning changes the operator average by
+[
+ Pr(V>256)lefrac{mathbb E(2^V)}bbE(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}):
+cite[Proposition~1]{FoxGlynn1988}):
 
- \norm{P(W)-E}<2^{-116}.
+ norm{P(W)-E}<2^{-116}.
 
 
-Put $Q=2^{56}$, $q_j=\lfloor Qp_j\rfloor$.  From $z=0$, for
+Put $Q=2^{56}$, $q_j=lfloor Qp_jrfloor$.  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 $u+v\omega$).  The ideal suffix at step $j$ is
-$Q\sum_{l=j}^{256}p_lW^{l-j}e_r$ of norm at most $Q$, and one
+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.
+ 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).
@@ -1213,39 +1213,39 @@ 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\cdot10^{-7},
+ e^{x_n/2}√(3^n)(frac{257(2√(3^n)+1)}Q+2^{-116})
+ <4cdot10^{-7},
 
-because $√(3^n)\le243$, $x_n/2<6$, $e^{x_n/2}<3^6$, and
-$3^6\cdot243(125159/2^{56}+2^{-116})<4\cdot10^{-7}$.
+because $√(3^n)le243$, $x_n/2<6$, $e^{x_n/2}<3^6$, and
+$3^6cdot243(125159/2^{56}+2^{-116})<4cdot10^{-7}$.
 
 ## 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).
+$pack(y)=sum_{t=0}^{w-1}y_t2^{64t}$ (arbitrary precision).
 Integer linear arithmetic preserves this identity before the floor even
-with carries.  For either coefficient, $|u|,|v|\le2|u+v\omega|$, so by
+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
+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 $pack((\lfloor y_t/16\rfloor)_t)$.  Adding the
+[
+ texttt{(((Y+bias)>{}>4)&mask)-bias16}
+]
+equals $pack((lfloor y_t/16rfloor)_t)$.  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
-$\lfloor(y_t+2^{63})/16\rfloor$ in lane $t$; subtracting $2^{59}$ gives
-$\lfloor y_t/16\rfloor$, also for negative $y_t$.  Hence the packed
+$lfloor(y_t+2^{63})/16rfloor$ in lane $t$; subtracting $2^{59}$ gives
+$lfloor y_t/16rfloor$, 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 $u+v\omega$ the squared modulus is $u^2-uv+v^2$, and summing
+exactly; for $u+vomega$ the squared modulus is $u^2-uv+v^2$, and summing
 these integers with representative multiplicities gives the integer
 $t_{n,g}$.
 
@@ -1254,122 +1254,122 @@ $t_{n,g}$.
 The integer recurrence gives:
 
 {ccl}
-\toprule
-$n$&$g$&$t_{n,g}$\\
-\midrule
-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\\
-\bottomrule
+toprule
+$n$&$g$&$t_{n,g}$
+midrule
+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
+bottomrule
 
 
 Enclose the scalar factor by
-\[
- L_n=\sum_{j=0}^{100}(x_n^j)/(j!),\qquad
- U_n=L_n+2\frac{x_n^{101}}{101!}.
-\]
+[
+ L_n=sum_{j=0}^{100}(x_n^j)/(j!),qquad
+ 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\cdot10^{-6}
- <\frac{L_nt_{n,g}}{Q^2}
- <\frac{U_nt_{n,g}}{Q^2}
- <c_{n,g}+2\cdot10^{-6}.
+ c_{n,g}-2cdot10^{-6}
+ <frac{L_nt_{n,g}}{Q^2}
+ <frac{U_nt_{n,g}}{Q^2}
+ <c_{n,g}+2cdot10^{-6}.
 
 These put every scaled norm below $4$, so by
 (eq:thermal120-scaled-error) replacing its square by the true trace
-costs less than $(8+4\cdot10^{-7})4\cdot10^{-7}$.  Together with
+costs less than $(8+4cdot10^{-7})4cdot10^{-7}$.  Together with
 (eq:thermal120-rational-enclosure),
-\[
+[
  |Z_n(21/2,g)-c_{n,g}|
- <2\cdot10^{-6}+(8+4\cdot10^{-7})4\cdot10^{-7}<10^{-5}.
-\]
+ <2cdot10^{-6}+(8+4cdot10^{-7})4cdot10^{-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
-\[
- b=(49)/(4),\qquad J=280,\qquad Q=2^{55},\qquad 4\le n\le12,
-\]
-with twists the identity and rotation by $\pi$ about $z$ (the latter has
+[
+ b=(49)/(4),qquad J=280,qquad Q=2^{55},qquad 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 $w=(w_0,...,w_{n-1})$, $w_i\in\{-1,0,1\}$,
+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),\qquad D=32R ,
+ R=I-(frac32nI+H)/(16),qquad D=32R ,
 
 an integer matrix with diagonal
-$D_{w,w}=32-3n-2\sum_{i=0}^{n-1}w_{i-1}w_i$, an entry $-2$ for an allowed
+$D_{w,w}=32-3n-2sum_{i=0}^{n-1}w_{i-1}w_i$, 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
-\[
- 0\le\tfrac32nI+H\le\tfrac52nI,\qquad
- \spec(R)⊆[1-5n/32,1]⊆[-\tfrac78,1],
-\]
+[
+ 0letfrac32nI+Hletfrac52nI,qquad
+ spec(R)⊆[1-5n/32,1]⊆[-tfrac78,1],
+]
 so $R$ is a self-adjoint contraction and
-\[
- F=\exp\![-\tfrac b2(\tfrac32nI+H)]=\exp[98(R-I)] .
-\]
+[
+ F=exp![-tfrac b2(tfrac32nI+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,\qquad x_n=bn(\tfrac32-a).
+ Z_n(b,g)=e^{x_n}norm{F}_F^2,qquad x_n=bn(tfrac32-a).
 
 Define the conditioned Poisson probabilities
 
- p_j=\frac{v_j}{\sum_{\ell=0}^Jv_\ell},\quad
- v_j=98^j(J!)/(j!),\quad 0\le j\le J .
+ p_j=frac{v_j}{sum_{ell=0}^Jv_ell},quad
+ v_j=98^j(J!)/(j!),quad 0le jle J .
 
-Since $\norm R\le1$, discarding the Poisson tail and renormalizing
+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 $2^j\ge2^{281}$ in the tail and $e^{-98}\sum_{j\ge0}196^j/j!=e^{98}$,
+[
+ 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←\Bigl\lfloor(DX)/(32)\Bigr\rfloor+c_je_r,\qquad
- c_j=\lfloor Qp_j\rfloor,
+ X←Bigllfloor(DX)/(32)Bigrrfloor+c_je_r,qquad
+ c_j=lfloor Qp_jrfloor,
 
 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 $Y_j=Q\sum_{\ell=j}^Jp_\ell R^{\ell-j}e_r$,
-$\norm{Y_j}_2\le Q$.  In a block of dimension $d_0$ a vector floor adds
+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=\lfloor√(3^n)\rfloor+1,\quad
- E_n=281(d_n+1)+1,\quad \mathcal E_n=d_nE_n,
+ d_n=lfloor√(3^n)rfloor+1,quad
+ E_n=281(d_n+1)+1,quad mathcal E_n=d_nE_n,calE_n=d_nE_n,
 
 every computed column obeys
 
- \norm{X-QFe_r}_2\le E_n .
+ 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 $d_n\le730$; any row of $D$ has absolute
-sum at most $|32-3n|+2n+4n\le76$.  Hence every product and partial sum is
-below $76\cdot2Q=5476377146882523136<2^{63}-1$, and division by $32$ plus
-$c_j\le Q$ stays below $207165582859042816<2^{63}-1$.  Starting at $X=0$
+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 $76cdot2Q=5476377146882523136<2^{63}-1$, 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
@@ -1382,16 +1382,16 @@ 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 $\sum_r\nu_r=3^n$.
+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 $(k-1,k)$, $k=-q\bmod n$, and
+the seam; left rotation by $q$ moves it to $(k-1,k)$, $k=-qbmod n$, and
 the diagonal unitary
-\[
- G_k|w>=(-1)^{\sum_{j=0}^{k-1}w_j}|w>,\quad G_0=I,
-\]
+[
+ G_k|w>=(-1)^{sum_{j=0}^{k-1}w_j}|w>,quad 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
@@ -1400,14 +1400,14 @@ untwisted chain needs no correction.
 Let $X_r$ be the output of (eq:thermal72-horner) at representative
 $r$ and
 
- W_{n,g}=\sum_r\nu_r\sum_wX_r(w)^2 .
+ 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 $Q^2\norm{F}_F^2$, the second $W_{n,g}$.
+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}}-Q\norm{F}_F|\le√(3^n)E_n<\mathcal E_n,
+ |sqrt{W_{n,g}}-Qnorm{F}_F|le√(3^n)E_n<mathcal E_n,calE_n,
 
 which proves the full trace estimate without equivariance of the rounded
 columns.
@@ -1416,48 +1416,48 @@ columns.
 
 Equations (eq:thermal72-coefficients),
 (eq:thermal72-horner), (eq:thermal72-total) give (second
-column: no twist; third: $\pi$ twist):
+column: no twist; third: $pi$ twist):
 
-\begingroup
-\footnotesize
+begingroup
+footnotesize
 
 {r|rr}
-\(n\)&\(W_{n,I}\)&\(W_{n,P}\)\\ \hline
-4&1298092848946206529211955779616040&683496640248050713756440676071\\
-5&9095379218684502088743241789&9205948219074287936685289758037\\
-6&11971284263768914176431179720622&266573596119930013833919176645\\
-7&15044150708248976451784593468&534330762492192297990602166370\\
-8&386135473475360677201628357188&42459883501386986922766507327\\
-9&4866417737994347551816846903&37716485750353424806876104310\\
-10&19772960374215472827775892486&5030484777222468978572235564\\
-11&841347687722481981930750335&2928767918985354021900689432\\
+(n)&(W_{n,I})&(W_{n,P}) hline
+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
+endgroup
 
 
 Turn these integers into the $b=49/4$ table.  Let
-\[
- r_{n,g}=\lfloor\sqrt{W_{n,g}}\rfloor,\quad
- s_n=\sum_{j=0}^{100}(x_n^j)/(j!),\quad
- t_n=\frac{x_n^{100}}{100!}.
-\]
-Since $0\le x_n<15$, later omitted terms have ratio at most $15/101$ and
-the tail is at most $15t_n/86<t_n$, so $s_n\le e^{x_n}\le s_n+t_n$.
+[
+ r_{n,g}=lfloorsqrt{W_{n,g}}rfloor,quad
+ s_n=sum_{j=0}^{100}(x_n^j)/(j!),quad
+ 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_n\max(0,r_{n,g}-\mathcal E_n)^2}{Q^2},\quad
- U_{n,g}=\frac{(s_n+t_n)(r_{n,g}+1+\mathcal E_n)^2}{Q^2},\quad
- L_{n,g}\le Z_n(b,g)\le U_{n,g},
+ L_{n,g}=frac{s_nmax(0,r_{n,g}-mathcal E_n)^2}calE_n)^2{Q^2},quad
+ U_{n,g}=frac{(s_n+t_n)(r_{n,g}+1+mathcal E_n)^2}calE_n)^2{Q^2},quad
+ L_{n,g}le Z_n(b,g)le U_{n,g},
 
 all rational/integer with the integer square root fixed by
-$r^2\le W<(r+1)^2$.  For the integer centers $C_{n,g}$ of the $b=49/4$
+$r^2le W<(r+1)^2$.  For the integer centers $C_{n,g}$ of the $b=49/4$
 table, direct evaluation yields
-\[
- L_{n,g}>\frac{C_{n,g}-30}{10^6},\qquad
- U_{n,g}<\frac{C_{n,g}+30}{10^6}\qquad(4\le n\le12,\ g=I,P),
-\]
+[
+ L_{n,g}>frac{C_{n,g}-30}{10^6},qquad
+ U_{n,g}<frac{C_{n,g}+30}{10^6}qquad(4le nle12, g=I,P),
+]
 each strict inequality with margin above $13/10^6$.  This gives a
 reproducible finite verification of all eighteen enclosures and proves the
 $b=49/4$ table.
@@ -1472,18 +1472,18 @@ powers of integer matrices of order at most $162$.
 
 ## Norm and energy contractions
 
-For $N\ge4$ and real $D\times D$ matrices $A_s$ ($s=-1,0,1$) define, in
+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}}),
+ 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
+cite[Section~2.1, Equations~(2)--(3)]{PerezGarcia2007}.  Put
 
- B_{st}&=A_sA_t,\quad T=\sum_{s=-1}^1A_s ⊗  A_s,
- \\
- O&=\sum_{s,t=-1}^1st\,B_{st} ⊗  B_{st}
- +\sum_{\substack{s=-1,0\\t=0,1}}
+ B_{st}&=A_sA_t,quad 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}).
  
@@ -1491,84 +1491,84 @@ the periodic matrix-product representation of
 All traces are ordinary, unnormalized.
 
 
-Let $N\ge4$, $D\ge1$ and $A_{-1},A_0,A_1\in M_D(\R)$; define
-$\psi_N,T,O$ by (eq:trial-vector)--(eq:trial-insertion) and,
+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
-$<\psi_N,H_N\psi_N>=NU_N$.  In particular, if $V_N>0$ and
+$V_N=Tr(T^N), U_N=Tr(T^{N-2}O)$.  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
+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
+$sqrt2$).  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
+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
+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 $P_{\alpha\beta}$ of Table~tab:trial-pairs,
-defined only when $\ell_\beta=\ell_\alpha+2$.
+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
+centering
 {c|rrrrrrrr}
-$\alpha$&1&2&3&4&5&6&7&8\\ \hline
-$D_\alpha$&$-289$&$-84$&$-9$&33&$-22$&27&71&$-11$
+$alpha$&1&2&3&4&5&6&7&8 hline
+$D_alpha$&$-289$&$-84$&$-9$&33&$-22$&27&71&$-11$
 
 Caption: Diagonal parameters, common to both lengths.
 
 
 
 [ht]
-\centering
-{c|r@{\qquad}c|r@{\qquad}c|r}
-$(\alpha,\beta)$&$P_{\alpha\beta}$&
-$(\alpha,\beta)$&$P_{\alpha\beta}$&
-$(\alpha,\beta)$&$P_{\alpha\beta}$\\ \hline
-$(1,5)$&$-11$&$(2,7)$&$-49$&$(4,6)$&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\\
+centering
+{c|r@{qquad}c|r@{qquad}c|r}
+$(alpha,beta)$&$P_{alphabeta}$&
+$(alpha,beta)$&$P_{alphabeta}$&
+$(alpha,beta)$&$P_{alphabeta}$ hline
+$(1,5)$&$-11$&$(2,7)$&$-49$&$(4,6)$&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 $2$.
+Caption: Parameters joining virtual types whose $ell$ differ by $2$.
 
 
 
-Entry rules.  Row $(\alpha,d)$, column $(\beta,e)$, with $\ell=\ell_\alpha$,
-$k=\ell_\beta$, $s\in\{-1,0,1\}$.  The entry is zero unless
+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,\qquad |k-\ell|\le2,\qquad (k=\ell⇒\alpha=\beta).
+ e=d+2s,qquad |k-ell|le2,qquad (k=ell⇒alpha=beta).
 
 For an allowed entry set
-\[
- p=D_\alpha,& k=\ell,\\ P_{\alpha\beta},& k=\ell+2,\\
- -P_{\beta\alpha},& k=\ell-2,
-\]
-$r=(d+\ell)/2$, and the entry $pf$ where
+[
+ 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\\ \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
+ &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
 
 
 lists the values of $f$.  These rules do not depend on $N$.
@@ -1577,14 +1577,14 @@ lists the values of $f$.  These rules do not depend on $N$.
 
 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 $d-e$
+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\\ \hline
-\dim&1&8&32&80&136&162&136&80&32&8&1
+d-e&-10&-8&-6&-4&-2&0&2&4&6&8&10 hline
+dim&1&8&32&80&136&162&136&80&32&8&1
 ,
 
 summing to $676=26^2$, each block once.
@@ -1592,25 +1592,25 @@ 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
-\[
- P_0=I,\quad P_{2j}=P_j^2\ (j\ge1),\quad P_{2j+1}=P_j^2T_\delta\ (j\ge0),
-\]
+difference block giving $T_delta,O_delta$; compute powers by
+[
+ P_0=I,quad P_{2j}=P_j^2 (jge1),quad P_{2j+1}=P_j^2T_delta (jge0),
+]
 then accumulate
 
- U_N=\sum_\delta\sum_{i,j}(P_{N-2})_{ij}(O_\delta)_{ji},\quad
- V_N=\sum_\delta\sum_{i,j}(P_{N-2})_{ij}(T_\delta^2)_{ji},
+ U_N=sum_deltasum_{i,j}(P_{N-2})_{ij}(O_delta)_{ji},quad
+ 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,\qquad -c_NV_N\le10^{12}U_N<(-c_N+1)V_N,
- \qquad
+ V_N>0,qquad -c_NV_Nle10^{12}U_N<(-c_N+1)V_N,
+ qquad
  {c|r}
- N&c_N\\ \hline
- 72&1401482112601\\
+ N&c_N hline
+ 72&1401482112601
  120&1401482105174
  .
 
@@ -1634,29 +1634,29 @@ 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).
+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).