diff --git a/spin_one_periodic_haldane_v2.txt b/spin_one_periodic_haldane_v2.txt
index b4bb7ac..f93564e 100644
--- a/spin_one_periodic_haldane_v2.txt
+++ b/spin_one_periodic_haldane_v2.txt
@@ -124,7 +124,7 @@ 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_jbfS_jcdotmathbf S_bfS_{j+1}, whereas
+to a scalar, with h=mathbf S_jbfS_j·mathbf 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
@@ -248,7 +248,7 @@ We need the diagonal rotations
  P_z=diag(-1,-1,1),
 ]
 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
+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
@@ -313,8 +313,8 @@ 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 GcalGsubset SO(3) be the group of proper signed permutation
-matrices (it contains D_2 and C).  For ginmathcal GcalG write
+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}.
@@ -325,11 +325,11 @@ Rotation covariance of the Cartesian matrices gives
 
 
 [Spatial transfer]
-For each b>0 there are a Hilbert space mathcal K_b,calK_b, a compact
+For each b>0 there are a Hilbert space K_b, a compact
 self-adjoint operator X_b, and a unitary representation
-gmapsto U_g of mathcal GcalG commuting with X_b, such that
+g|-> U_g of G commuting with X_b, such that
 
- Z_n(b,g)=Tr(X_b^nU_g)(nge4, ginmathcal G).calG).
+ 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
@@ -352,7 +352,7 @@ 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},
-  mathcal K_b=L^2(Ω_b,calK_b=L^2(Ω_b,mu;C).
+  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
@@ -436,7 +436,7 @@ 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_nqquad(nge4).
+ 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
@@ -711,7 +711,7 @@ 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 30cdot10^{-6} at b=49/4:
+b=21/2 and below 30·10^{-6} at b=49/4:
 [
 {c|rrr}
  &multicolumn{3}{c}{b=21/2}
@@ -753,16 +753,16 @@ Two elementary lemmas convert the tables into spectral bounds.
 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,calC(m,h,r)=qh^r+(m-qh)^r, q=lfloor m/hrfloor.
+ C(m,h,r)=qh^r+(m-qh)^r,calC(m,h,r)=qh^r+(m-qh)^r, q=floor( m/h).
 
-Then sum_it_i^rlemathcal C(m,h,r);calC(m,h,r); without the cap,
+Then sum_it_i^rleC(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 xmapsto x^r shows the sum of rth powers does not
+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
@@ -805,8 +805,8 @@ Work at b_0=21/2.  From the b_0=21/2 table and
  m_I=(1042655)/(10^6), m_O=(40041)/(10^6),
  m_N=(401625)/(10^6),
 ]
-with margins 19/(12cdot10^6), 7/(12cdot10^6),
-3/(4cdot10^6) over centers plus 10^{-5}.  Apply
+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 .
@@ -876,9 +876,9 @@ 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)inmathcal V_t}calV_t
+ R(t)=max_{(x,y)inV_t}
  (x^2+(y^2)/(2)+((D-t-x-y)^2)/(3)),
- mathcal V_t=calV_t={(0,0),(tau-t,0),(tau-t,tau_0-tau),(0,tau_0-t)}.
+ 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
@@ -919,7 +919,7 @@ spatial weights used for S.
 At inverse temperature b=784,
 [
  1-S(2304,b)le(84513)/(10^7)<frac1{100},
- 1-T(4608,b)le(27493)/(5cdot10^6)<frac1{100}.
+ 1-T(4608,b)le(27493)/(5·10^6)<frac1{100}.
 ]
 
 
@@ -928,7 +928,7 @@ 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=30cdot10^{-6}.
+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,
@@ -960,12 +960,12 @@ 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)=3mathcal C(m_N,v_N^calC(m_N,v_N^{12},k/12),
+ B(k)=3C(m_N,v_N^calC(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)lemathcal C(m_J,calC(m_J,ell^{12},6)+B(72)<1, with strict margin
+Z_{72}(b_0)leC(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
@@ -1003,7 +1003,7 @@ Exact rational evaluation gives
 ]
 Each update lies in [0,1), where f increases, so upward rounding
 preserves the bounds.  Length and inverse temperature have multiplied by
-2^6: 36cdot64=2304 and (49/4)cdot64=784.  The last row proves
+2^6: 36·64=2304 and (49/4)·64=784.  The last row proves
 the proposition.
 
 
@@ -1084,7 +1084,7 @@ 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
+[H,A]=sum_{jinZ}[mathbf S_jbfS_j·mathbf S_bfS_{j+1},A], only finitely
 many terms nonzero.
 
 
@@ -1155,7 +1155,7 @@ 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>mapsto e^{-itheta x_{n-1}}|s(x)> commutes with H.
+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
@@ -1183,7 +1183,7 @@ let
 These are mean-84 Poisson probabilities conditioned to le256; for
 a mean-84 Poisson variable V,
 [
- Pr(V>256)lefrac{mathbb E(2^V)}bbE(2^V){2^{257}}=frac{e^{84}}{2^{257}}
+ 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
@@ -1193,7 +1193,7 @@ cite[Proposition~1]{FoxGlynn1988}):
  norm{P(W)-E}<2^{-116}.
 
 
-Put Q=2^{56}, q_j=lfloor Qp_jrfloor.  From z=0, for
+Put Q=2^{56}, q_j=floor( Qp_j).  From z=0, for
 j=256,...,0:
 
  z←fl(Bz/16)+q_je_r ,
@@ -1214,10 +1214,10 @@ 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})
- <4cdot10^{-7},
+ <4·10^{-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}.
+3^6·243(125159/2^{56}+2^{-116})<4·10^{-7}.
 
 ## Exact evaluation by packed integers
 
@@ -1237,12 +1237,12 @@ bias16=2^{59}s_w.  For Y=pack(y) obeying
 [
  texttt{(((Y+bias)>{}>4)&mask)-bias16}
 ]
-equals pack((lfloor y_t/16rfloor)_t).  Adding the
+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
-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
+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
@@ -1278,18 +1278,18 @@ 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}-2cdot10^{-6}
+ c_{n,g}-2·10^{-6}
  <frac{L_nt_{n,g}}{Q^2}
  <frac{U_nt_{n,g}}{Q^2}
- <c_{n,g}+2cdot10^{-6}.
+ <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+4cdot10^{-7})4cdot10^{-7}.  Together with
+costs less than (8+4·10^{-7})4·10^{-7}.  Together with
 (eq:thermal120-rational-enclosure),
 [
  |Z_n(21/2,g)-c_{n,g}|
- <2cdot10^{-6}+(8+4cdot10^{-7})4cdot10^{-7}<10^{-5}.
+ <2·10^{-6}+(8+4·10^{-7})4·10^{-7}<10^{-5}.
 ]
 This proves the b=21/2 table.
 
@@ -1346,8 +1346,8 @@ then e<4.
 
 For a unit basis column e_r, start at X=0 and for j=J,...,0 do
 
- X←Bigllfloor(DX)/(32)Bigrrfloor+c_je_r,
- c_j=lfloor Qp_jrfloor,
+ X←Biglfloor((DX)/(32)Bigr)+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
@@ -1356,8 +1356,8 @@ 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,
- E_n=281(d_n+1)+1, mathcal E_n=d_nE_n,calE_n=d_nE_n,
+ d_n=floor(√(3^n))+1,
+ E_n=281(d_n+1)+1, E_n=d_nE_n,
 
 every computed column obeys
 
@@ -1368,7 +1368,7 @@ 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 76cdot2Q=5476377146882523136<2^{63}-1, and division by 32 plus
+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
@@ -1407,7 +1407,7 @@ 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<mathcal E_n,calE_n,
+ |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.
@@ -1438,7 +1438,7 @@ endgroup
 
 Turn these integers into the b=49/4 table.  Let
 [
- r_{n,g}=lfloorsqrt{W_{n,g}}rfloor,
+ 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!}.
 ]
@@ -1447,8 +1447,8 @@ 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}-mathcal E_n)^2}calE_n)^2{Q^2},
- U_{n,g}=frac{(s_n+t_n)(r_{n,g}+1+mathcal E_n)^2}calE_n)^2{Q^2},
+ 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