diff --git a/uniform_trace_reconstruction_v2.txt b/uniform_trace_reconstruction_v2.txt
index dc059c5..9caba79 100644
--- a/uniform_trace_reconstruction_v2.txt
+++ b/uniform_trace_reconstruction_v2.txt
@@ -138,7 +138,7 @@ Servedio, and separately Nazarov and Peres, proved
 exp(O(n^{1/3})) bounds together with matching limitations for
 methods that use only trace means. Their complex-analytic approach ties
 deletion statistics to bounded-coefficient polynomials, with analytic
-inequalities of Borwein and Erd'elyi in the background. That
+ineualities of Borwein and Erd'elyi in the background. That
 mean-based lower bound restricts a method, not arbitrary trace
 algorithms.
 
@@ -384,7 +384,7 @@ exponentially damped Fourier sum
 Every monomial in f_j uses positions at most j, so
 Delta[f_j]=0 for j<d; hence, despite the exponent being centred at
 d, only jge d contribute. For the base case k=1, where f_j=X_j,
-this reads S_f(s,0)=calS_f(s,0)=sum_{jge d}Delta[X_j],e^{-s(j-d)}, whose
+this reads S_f(s,0)=S_f(s,0)=sum_{jge d}Delta[X_j],e^{-s(j-d)}, whose
 first term is Delta[X_d]=pm1; that single term is what makes the
 starting signal about 1/2 at decay s=4, as made precise in
 (eq:initial-signal). The whole multiscale argument has one goal: to preserve a detectable signal
@@ -483,18 +483,18 @@ with C absolute and Uge1. By prefix cancellation,
 ]
 First, P is shown to keep a signal at almost the original decay:
 
- abs{P(e^{-s_1+iiomega})}getfrac12 e^{-E}
+ abs{P(e^{-s_1+iiomega})}ge1/2 e^{-E}
  quadfor somequad abs{s_1/s-1}le10^{-4}.
 
 
 Let V be a centred real normal variable with variance sigma^{-2}.
-Since W(l),mathbf EbfE,e^{Vl}=1,
+Since W(l),E,e^{Vl}=1,
 [
- S_f(s,omega)=mathbf EbfE,P(e^{-s+V+iiomega}).
+ S_f(s,omega)=E,P(e^{-s+V+iiomega}).
 ]
 Write delta=10^{-4}. Exponential tilting gives
 [
- W(l),mathbf E[e^bfE[e^{Vl}mathbf 1_bf1_{{abs V>delta s}}]
+ W(l),E[e^E[e^{Vl}1_bf1_{{abs V>delta s}}]
  =Pr!(abs{N(l/sigma^2,sigma^{-2})}>delta s).
 ]
 For 0le lledelta ssigma^2/2 this probability is at most
@@ -531,7 +531,7 @@ Poisson measure at e^{-s_1+iiomega}.
 To apply it, factor the nonzero polynomial P into linear factors. A
 factor with a root outside the unit disk has harmonic logarithmic
 magnitude inside. For a root a inside the disk, the identity
-abs{e^{ii t}-a}=abs{1-overlineoverline aa e^{ii t}} and the inequality
+abs{e^{ii t}-a}=abs{1-overlineoverline aa e^{ii t}} and the ineuality
 [
  abs{1-overlineoverline aa z}^2-abs{z-a}^2=(1-abs a^2)(1-abs z^2)ge0
 ]
@@ -734,7 +734,7 @@ If lambda=0, use two endpoints and J=1. For each interior y_i
 choose a pair from (eq:paired-frequencies) and set
 [
  V_0=1,
- V_i=frac{A(t)A(t')bar}{B_x(t)B_x(t')bar} (0<i<J),
+ V_i=A(t)A(t')bar/B_x(t)B_x(t')bar (0<i<J),
   V_J=(A(lambda))/(B_x(lambda)).
 ]
 The denominators are nonzero by
@@ -758,7 +758,7 @@ Since v_0=1 has norm one, Lemma~lem:gram-chain yields
  =abs{B_x(lambda)},abs{< v_J,v_0>-1}
  le2UJ^2e^{-2.44E}<e^{-2.40E}.
 ]
-The last strict inequality holds after enlarging C_0, because
+The last strict ineuality holds after enlarging C_0, because
 log(2UJ^2)<0.04E by (eq:multiscale-cost). This contradicts
 (eq:single-frequency) and proves the lemma.
 
@@ -796,7 +796,7 @@ dampings unchanged.
 {itshape A Gaussian contour with controlled total frequency.}
 Recall sigma=Ksqrt E/S' and b=beta S'. For every real l,
 
- W(l)=frac{sigma}{ii√(2pi)}
+ W(l)=sigma/ii√(2pi)
        int_{-ii∞}^{ii∞}
        exp((sigma^2v^2)/(2)+vl),dv.
 
@@ -834,7 +834,7 @@ Let nu be the resulting complex measure on RR^m after removing the
 position-dependent factors. Outside the measure-zero set Ω=0,
 
  dnu(u)=
- (frac{sigma}{√(2pi)})^m
+ (sigma/√(2pi))^m
  exp((sigma^2)/(2)sum_{r=1}^m v_r^2)
  (1+iirhosgnΩ),du.
 
@@ -843,7 +843,7 @@ ii,du_m. This quantity is bounded uniformly over the frequencies in
 (eq:endpoint-window); the bounds also give absolute convergence.
 
 Put delta_0=1-4rho^2>0 and c_1=delta_0/2. Since mle4 and
-|sum t_r|le b, Young's inequality gives an absolute C_4 with
+|sum t_r|le b, Young's ineuality gives an absolute C_4 with
 
  sum_{r=1}^m(re v_r)^2
  &=(m-1)b^2+(mb+rho|Ω|)^2 notag
@@ -852,7 +852,7 @@ Put delta_0=1-4rho^2>0 and c_1=delta_0/2. Since mle4 and
  &le (1-c_1)sum_r u_r^2+C_4b^2.
  
 
-The strict inequality 4rho^2<1 is essential: summing at most four
+The strict ineuality 4rho^2<1 is essential: summing at most four
 imaginary coordinates can multiply their squared norm by four, and the
 sector constant 2.002 leaves a positive Gaussian margin. For example
 C_4=3+(4+rho)^2(1+8rho^2/delta_0) works. With
@@ -947,7 +947,7 @@ interleaving is fixed, choosing arbitrary strictly increasing values for
 its v distinct positions specifies exactly one assignment to its
 labeled tuples, so no further restriction remains on the new tuple. Its
 largest position is i_v=j_m. Summing all these assignments gives an
-order-v feature f^{I,u}calI,u with outer weight
+order-v feature f^{I,u}I,u with outer weight
 exp((-s'+iiΩ)(i_v-d)).
 
 The needed combinatorial fact controls its height.
@@ -1014,13 +1014,13 @@ is the sum of a_e over edges crossing it, hence at least that of its
 assigned edge. Since 1+1/a decreases with a,
 Lemma~lem:edge-gap-assignment gives
 
- H_{f^{I,u}calI,u}
+ H_{f^{I,u}I,u}
  &leprod_{e assigned}(1+1/a_e)
  leprod_{e present}(1+1/a_e) notag
  &le H_f^m(1+1/b)^{m-1}le H_f^4(1+1/b)^3.
  
 
-The second inequality uses that all factors are at least one; the
+The second ineuality uses that all factors are at least one; the
 endpoint-link dampings are rbge b. This covers all overlaps between
 old tuples and also the single-distinct-position case, where the height
 is the empty product one.
@@ -1032,15 +1032,15 @@ corresponding measure,
 
  Delta[product]
  =sum_{I}{RR^m}
-   S_{f^{I,u}calI,u}(s',Ω),dnu_{I}
+   S_{f^{I,u}I,u}(s',Ω),dnu_{I}
 
 All estimates are uniform over interleavings, and the integrands are
 uniformly bounded even outside U. Indeed, for each fixed
-endpoint j the coefficients of f_j^{I,u}calI,u have absolute sum at
+endpoint j the coefficients of f_j^{I,u}I,u have absolute sum at
 most the height in (eq:interleaving-height), terms with j<d
 vanish under Delta by prefix agreement, and s'ge b gives
 
- |S_{f^{I,u}calI,u}(s',Ω)|
+ |S_{f^{I,u}I,u}(s',Ω)|
  le 2H_f^4(1+1/b)^3sum_{lge0}e^{-bl}
  le 2H_f^4(1+1/b)^4.
 
@@ -1121,7 +1121,7 @@ range gives
   t_hleθ^{,2^{h-J_*}-1}
   leexp{-Q(2^{h-J_*}-1)}le n^{-1}, S_hle n^{-1}.
 
-The last inequality uses 2^{h-J_*}ge8Y and Q(Y-1)=log n. At each
+The last ineuality uses 2^{h-J_*}ge8Y and Q(Y-1)=log n. At each
 transition 0<S_jlemin{1,eta_0S_{j-1}}, as the proposition requires.
 
 The cost of these small scales is also needed. Put A_0=Q+log Y. With
@@ -1241,7 +1241,7 @@ angles gives
 If Uge1 or Vge1 the bracket is at least one; otherwise
 U+V^2ge U^2+V^2ge(U+V)^2/2. Since abspsile U+V, in all cases
 [
-  Tgetfrac12gammamin{1,psi^2}.
+  Tge1/2gammamin{1,psi^2}.
 ]
 For 0<Tle1 this gives
 [
@@ -1357,7 +1357,7 @@ A common real value of v moves every starting coordinate into the trace
 half-plane at once. Concretely, in the Z-plane the region
 abs{Z-q}le p is a disk; after the Cayley map
 w=(1+Z)/(1-Z) it becomes the half-plane re wge q/p. Each path
-w_r(v)=(1+v)re w_r+ii,{rm Im},w_r is a horizontal segment starting
+w_r(v)=(1+v)re w_r+ii,{Im},w_r is a horizontal segment starting
 at w_r and ending at w_r(R) with re w_r(R)=(1+R)xi_rge q/p. All
 coordinates use the same parameter 0le vle R; the proof interpolates
 F(Z_0(v),...,Z_{k-1}(v)) on one complex strip, simultaneously for
@@ -1535,7 +1535,7 @@ Assume for now p<1. Choose an integer D and a positive dyadic number
 zeta=2^{-b_zeta} so that
 
  D_*le Dle CY^2,
- zetaletfrac14,
+ zetale1/4,
  4zeta<e^{-C_{sep}B(n,p)},
  b_zeta=O(B(n,p)),
 
@@ -1543,22 +1543,22 @@ with C_{sep} from Theorem~thm:separation. Integer
 formulas for these choices come later. Let
 [
  I_D={I⊆{1,...,n}:1le|I|le D},
- N=|I_D|,calI_D|,
+ N=|I_D|,I_D|,
  T=512(1+N)zeta^{-2},
 ]
 which is an integer. From T independent traces form the empirical mean
-widehatwidehat t_It_I of the rank observable defining T_I, for every
+that_It_I of the rank observable defining T_I, for every
 IinI_D. These are ordinary trace observables: a requested
 rank that is absent contributes zero.
 
 For each fixed input x let G be the event
 
- |widehatwidehat t_I-T_I(x)|t_I-T_I(x)|lezeta/8quadfor every IinI_D.
+ |that_I-T_I(x)|t_I-T_I(x)|lezeta/8quadfor every IinI_D.
 
 Each observable takes values in {0,1}, so its empirical mean has
 variance at most 1/(4T). Chebyshev and a union bound give
 
- P_x( G^c)calG^c)le(16N)/(Tzeta^2)lefrac1{32}.
+ P_x( G^c)G^c)le(16N)/(Tzeta^2)lefrac1{32}.
 
 The same event controls all the prefix tests, even though the prefixes are
 chosen using the data.
@@ -1569,7 +1569,7 @@ moment coordinates form a real vector m. Consider the rational
 constraints
 
  min[-1,1]^l, M(m)succeq0,
- |EE_m[T_I]-widehatwidehat t_I|t_I|lezetaquad(IinI_D),
+ |EE_m[T_I]-that_I|t_I|lezetaquad(IinI_D),
 
 where l is the number of moment coordinates. Substituting the prefix and
 writing each remaining bit as one half of one plus its sign makes
@@ -1583,7 +1583,7 @@ prefix has a moment vector m^*in[-1,1]^l such that, with G the
 dimension of its moment matrix,
 
  M(m^*)succeq(zeta/8)Id_G,
- |EE_{m^*}[T_I]-widehatwidehat t_I|t_I|lezeta/2quad(IinI_D).
+ |EE_{m^*}[T_I]-that_I|t_I|lezeta/2quad(IinI_D).
 
 
 
@@ -1600,12 +1600,12 @@ its (I,J) entry is zero unless I=J. The mixture has the stated matrix
 lower bound and coordinates in [-1,1]. Both constituent distributions
 give channel expectations in [0,1], so mixing changes each T_I(x) by
 at most zeta/8; (eq:empirical-good) then bounds the discrepancy
-from widehatwidehat t_It_I by zeta/4, stronger than required.
+from that_It_I by zeta/4, stronger than required.
 
 
 The next lemma gives the bounded procedure for
 (eq:prefix-feasibility). It only promises to succeed when there is
-slack, and every reported solution satisfies the exact inequalities. This
+slack, and every reported solution satisfies the exact ineualities. This
 sidesteps deciding arbitrary semidefinite feasibility. The procedure is a
 projected relaxation method in the tradition of Agmon and
 Motzkin--Schoenberg; the iteration and its rounding analysis are given
@@ -1620,7 +1620,7 @@ nonempty sign moments. Let M(m) be their moment matrix, of dimension
 [
  G=sum_{j=0}^{min(D,r)}binom rj.
 ]
-Suppose a finite list of rational affine inequalities ell_t(m)le b_t
+Suppose a finite list of rational affine ineualities ell_t(m)le b_t
 is given, each with nonconstant coefficient vector of ell^1-norm at
 most one, together with a dyadic 0<zetale1/4. Then a deterministic
 rational procedure either returns a vector with
@@ -1640,14 +1640,14 @@ log(1/zeta). These time bounds hold without the slack assumption.
 
 
 Proof. 
-If l=0 the matrix is [1] and all inequalities are constant, so direct
+If l=0 the matrix is [1] and all ineualities are constant, so direct
 rational comparisons suffice. Suppose l>0. Exact separating
 directions, then an iteration with bounded precision.
 
 {itshape Directions from violated constraints.}
 At a current point m^{cur}in[-1,1]^l first check the affine
-inequalities and the matrix condition exactly. If all hold, return the
-current vector. A violated inequality ell_t(m^{cur})>b_t gives
+ineualities and the matrix condition exactly. If all hold, return the
+current vector. A violated ineuality ell_t(m^{cur})>b_t gives
 the direction a equal to minus its nonconstant coefficient vector. Then
 norm a_2le1 and, under the slack assumption,
 < m^*-m^{cur},a>gezeta/2gelambda.
@@ -1667,7 +1667,7 @@ Thus either violated condition yields a rational direction with
 
  norm a_2le1, < m^*-m^{cur},a>gelambda
 
-whenever a slack solution exists. A violated constant inequality has zero
+whenever a slack solution exists. A violated constant ineuality has zero
 direction; that cannot happen in the slack case, and otherwise the
 iteration is still bounded.
 
@@ -1733,7 +1733,7 @@ slack solution.
 
 
 For a prefix test, write each absolute-value constraint of
-(eq:prefix-feasibility) as two affine inequalities and apply
+(eq:prefix-feasibility) as two affine ineualities and apply
 Lemma~lem:rational-feasibility. Lemma~lem:prefix-slack
 verifies exactly the slack hypotheses for a correct prefix, and every
 acceptance, on any data, gives an actual moment system satisfying