diff --git a/goe_rrg_bulk_universality_v2.txt b/goe_rrg_bulk_universality_v2.txt
index 953d4ce..4632cb4 100644
--- a/goe_rrg_bulk_universality_v2.txt
+++ b/goe_rrg_bulk_universality_v2.txt
@@ -51,7 +51,7 @@ is a very sparse matrix, and it is real symmetric. Let
 be its eigenvalues, counted with multiplicity. They are real and lie in
 [-d,d].
 
-A dense ntimes n Gaussian matrix, by contrast, has all tfrac12 n(n+1)
+A dense ntimes n Gaussian matrix, by contrast, has all 1/2 n(n+1)
 entries random. Its eigenvalues are completely different objects at the
 macroscopic scale. And yet, as we now explain, near any fixed interior
 energy the two matrices produce the same local eigenvalue
@@ -63,7 +63,7 @@ of G_{n,d}: the empirical measure
 frac1nsum_{i=1}^ndelta_{lambda_i} converges weakly, in probability,
 to the Kesten--McKay density
 
-  rho_d(t);=;frac{d,√(4(d-1)-t^2)}{2pi,(d^2-t^2)}
+  rho_d(t);=;d,√(4(d-1)-t^2)/2pi,(d^2-t^2)
    for |t|<2√(d-1),
 
 and rho_d(t)=0 otherwise. It integrates to 1. Its support is the
@@ -232,9 +232,9 @@ Dyson Brownian motion universality, semicircle edge limits).
 edges. For fixed d,E and any fixed number of eigenvectors in a
 microscopic band, the joint values of those eigenvectors at random edge
 endpoints are Gaussian to polynomial precision.
-Mechanism: a colored-star counting inequality becomes a
-differential-entropy inequality after smoothing by a Gaussian field;
-then a strict tensor inequality turns the small entropy deficit into
+Mechanism: a colored-star counting ineuality becomes a
+differential-entropy ineuality after smoothing by a Gaussian field;
+then a strict tensor ineuality turns the small entropy deficit into
 moment matching, by induction over Hermite degree.
  the microscopic-limit step: endpoint resolvents and the
 microscopic limit. The moments upgrade to control of resolvent
@@ -310,9 +310,9 @@ subscript says otherwise we set, once and for all,
 
   rho=rho_d(E), p=frac Ed,
   a=(d)/(d-1)(1-p^2),
-  m_*=m_d(E+mathrm i0)=m_rmi0)=m_{rm r}+mathrm irmipirho .
+  m_*=m_d(E+i0)=m_rmi0)=m_{r}+irmipirho .
 
-Then rho>0, a>0, pin(-1,1), and m_{rm r} is a real number.
+Then rho>0, a>0, pin(-1,1), and m_{r} is a real number.
 Note that
 
   1-p^2=(d^2-E^2)/(d^2),
@@ -472,18 +472,18 @@ rescaling the spectral parameter and the resolvent by the fixed factor
 √(d-1) gives our normalization.
 
 We keep track of why their error has our shape. Choose fixed
-0<c<1, a_{rm HY}ge12, b_{rm HY}ge25a_{rm HY} and
-0<mathfrak r<c/32,frakr<c/32, and let r_{rm HY} be
-mathfrak rfrakrlog_{d-1}n rounded to an integer. Their Definition 1.1
+0<c<1, a_{HY}ge12, b_{HY}ge25a_{HY} and
+0<r<c/32,r<c/32, and let r_{HY} be
+rlog_{d-1}n rounded to an integer. Their Definition 1.1
 and Proposition 2.1 produce an event of probability tending to one on
 which all but a polynomially small fraction of vertices have a tree
 neighbourhood of radius (c/4)log_{d-1}n; that radius is much larger
-than r_{rm HY}, and in particular larger than any neighbourhood
+than r_{HY}, and in particular larger than any neighbourhood
 appearing below. On that event their resolvent comparison, with their
 error (4.6)--(4.7), bounds the discrepancy by a logarithmic factor
 times
 [
-  (d-1)^{-r_{rm HY}}+(nImzeta)^{-1/2}+(nImzeta)^{-2/3}.
+  (d-1)^{-r_{HY}}+(nImzeta)^{-1/2}+(nImzeta)^{-2/3}.
 ]
 Since S dominates every fixed power of log n, and after decreasing
 c_1 and increasing B, this is the stated bound, uniform in the
@@ -491,7 +491,7 @@ indicated range. The same two statements supply polynomial rigidity and
 polylogarithmic complete delocalization, which is the bound on
 eigenvector entries.
 
-For the count, apply the trace estimate at zeta=E+mathrm irmi S^2/n.
+For the count, apply the trace estimate at zeta=E+irmi S^2/n.
 Every eigenvalue with |lambda_i-E|le S^2/n contributes at least
 n/(2S^2) to the imaginary part of (x_i-zeta/n)^{-1}; the
 normalized imaginary trace is bounded by the estimator above, and the
@@ -546,13 +546,13 @@ Finally, with U(E)=intlog|t-E|rho_d(t),dd t,
 
   -U(E)=-frac12log d+(d-1)/(2)log a-frac d4log(1-p^2),
 
-and replacing E by E+mathrm irmieta changes the real part of the
+and replacing E by E+irmieta changes the real part of the
 logarithmic potential by O(eta).
 
 
 Proof. 
 The Green function. Let o be the root of the full tree and let
-F(zeta) be the diagonal entry of (zeta-A_{rm branch})^{-1} at the
+F(zeta) be the diagonal entry of (zeta-A_{branch})^{-1} at the
 root of a rooted tree in which the root has d-1 children, recursively.
 Each child subtree is again such a branch, so a Schur complement at the
 root gives
@@ -567,7 +567,7 @@ Schur complement over the d branches gives
 ]
 The left side is intrho_d(t)/(zeta-t)dd t=-m_d(zeta), which proves
 (eq:tree-green). Its boundary imaginary part is
-pi^{-1}Im m_d(t+mathrm i0)=rmi0)=rho_d(t): this follows by checking that
+pi^{-1}Im m_d(t+i0)=rmi0)=rho_d(t): this follows by checking that
 the boundary function is nonnegative, compactly supported on
 [-2√(d-1),2√(d-1)], and has total mass 1, the latter from
 m_d(zeta)sim-1/zeta at infinity and a contour argument.
@@ -624,7 +624,7 @@ zeta=q^{-1}+(d-1)q:
 matched at infinity using m_d(zeta)sim-1/zeta. At the boundary,
 [
   |q|=(d-1)^{-1/2},
-  |1-q^2|=frac{√(d^2-E^2)}{d-1} ,
+  |1-q^2|=√(d^2-E^2)/d-1 ,
 ]
 by evaluating the quadratic at zeta=E. Substituting into
 (eq:tree-potential) and using a=d(1-p^2)/(d-1) in the form
@@ -635,7 +635,7 @@ log(sqrt3/2).)
 Finally, t|->log|t-E| is bounded near E because rho_d is,
 and
 [
-  intlog|1-(mathrm irmieta)/(t-E)|rho_d(t),dd t
+  intlog|1-(irmieta)/(t-E)|rho_d(t),dd t
   =frac12intlog(1+(eta^2)/((t-E)^2))rho_d(t),dd t
   =O(eta),
 ]
@@ -658,10 +658,10 @@ n^{-c} is more than enough to beat any subpolynomial factor. What
 would not be enough is abstract weak convergence, which carries no rate.
 
 The strategy is the entropy method of Backhausz--Szegedy, made
-quantitative. A counting inequality for colourings of the graph
+quantitative. A counting ineuality for colourings of the graph
 (the coloured-star count) bounds the number of labelings with
 given local statistics. After a Gaussian smoothing, which turns the
-counting bound into a differential-entropy inequality, the deficit
+counting bound into a differential-entropy ineuality, the deficit
 between the entropy of the star observation and half that of the edge
 observation is small. The projection step removes the
 one nearly deterministic relation (the eigenvector equation), and
@@ -742,7 +742,7 @@ Let T be the distribution of the ordered star observation, symmetrized
 over the d! orders of the neighbours, and let Q be the distribution
 of the oriented edge observation. Then T and Q are determined by the
 labeling, and they are the only local information the counting
-inequality sees. The following is the coloured-star count of
+ineuality sees. The following is the coloured-star count of
 Backhausz--Szegedy, with the dependence on a growing alphabet kept
 explicit.
 
@@ -787,7 +787,7 @@ the probability is
   -(nd)/(2)H(Q)+O(M^2log n).
 ]
 (The computation: each mlog m term with m=ndQ_{jk} contributes
-Q_{jk}log Q_{jk} times nd, the -tfrac12 ndlog(nd) parts of the
+Q_{jk}log Q_{jk} times nd, the -1/2 ndlog(nd) parts of the
 numerators cancel the denominator, and the residual terms are
 O(M^2log n).)
 
@@ -799,7 +799,7 @@ ordered types compatible with a given symmetrized T is at most
 exp(O(M^{d+1}log n)), because a type is a probability
 vector on M^{d+1} ordered observations. Summing over these, and using
 (eq:type-count-hypothesis), changes only the sublinear error term.
-Markov's inequality together with a union bound over all pairs
+Markov's ineuality together with a union bound over all pairs
 (T,Q) upgrades the statement about expectations to a statement
 holding for all pairs simultaneously with probability tending to one:
 there are only exp(O(M^{d+1}log n)) pairs, and each has negligible
@@ -844,7 +844,7 @@ eta↓0. For 0leεle1 define the smoothed field
 where V is the eigenvector field V(v)=(u_{i(1)}(v),...,u_{i(ell)}(v)).
 The parameter eta regularizes the almost-deterministic eigenvector
 relation; the independent parameter ε measures how much of the
-eigenvector signal is kept. We keep the entropy inequality uniformly in
+eigenvector signal is kept. We keep the entropy ineuality uniformly in
 ε and later choose a small power of n^{-1} when extracting
 each fixed moment.
 
@@ -852,14 +852,14 @@ each fixed moment.
 On events of graphs of probability tending to one, uniformly over all
 choices of V and all 0leεle1,
 
-  H(F_{rm star})-frac d2H(F_{rm edge})
-   ge frac1nH(Y_{rm global})-O(n^{-c}),
+  H(F_{star})-frac d2H(F_{edge})
+   ge frac1nH(Y_{global})-O(n^{-c}),
 
-where Y_{rm global} is the full nell-dimensional Gaussian array.
+where Y_{global} is the full nell-dimensional Gaussian array.
 Moreover
 
-  frac1nH(Y_{rm global})
-  =ell(frac12logfrac{2pi{rm e}eta}{pirho}-U(E))
+  frac1nH(Y_{global})
+  =ell(frac12logfrac{2pi{e}eta}{pirho}-U(E))
    +O(n^{-c}),
 
 with U(E)=intlog|t-E|rho_d(t),dd t from
@@ -895,12 +895,12 @@ therefore has, after averaging over locations, an off-block Frobenius
 norm of at most
 [
   C((TrGamma^2)/(n^2))^{1/2}
-  le frac{C}{√(neta)},
+  le C/√(neta),
 ]
 because TrGamma^2le Cn/eta, and the smallest eigenvalue of a
 single-edge covariance is at least ceta (use |A|le d). After
 whitening the two marginal blocks, the total-variation distance from the
-product law is at most |{rm off,block}|/lambda_{min}le
+product law is at most |{off,block}|/lambda_{min}le
 Cn^{-1/2}eta^{-3/2}; for reference this follows by differentiating the
 Gaussian density along the segment joining the two covariance matrices.
 Consequently, for two independent uniform oriented edges and any cell of
@@ -919,9 +919,9 @@ cells, so Cauchy--Schwarz bounds the mean ell^1 deviation
   le (M^2· Cn^{-1/2+2alpha})^{1/2}
   =n^{2ellgamma-1/4+alpha+o(1)}=O(n^{-c}),
 ]
-because 4ellgamma+2alpha<tfrac14. For two probability vectors on a
+because 4ellgamma+2alpha<1/4. For two probability vectors on a
 set of size M^2 at ell^1 distance h, the entropies differ by at
-most hlog(M^2)+H_{rm bin}(h), where H_{rm bin} is the binary
+most hlog(M^2)+H_{bin}(h), where H_{bin} is the binary
 entropy; therefore
 
   EE H(Q)=H(EE Q)+O(n^{-c}).
@@ -933,7 +933,7 @@ spectrum in [ceta,C/eta], because Gamma does and because a
 deterministic shift does not change covariance. Unbounded binning of a
 Gaussian vector and the entropy bound on a cube give
 [
-  frac1nH(L) ge frac1nH(Y_{rm global})
+  frac1nH(L) ge frac1nH(Y_{global})
   -elllog b-O(n^{-c});
 ]
 the clipping error is in fact superpolynomially small, since
@@ -943,7 +943,7 @@ superpolynomially small, uniformly in the shift. For either local
 observation, of dimension ell(d+1) for the star and 2ell for the
 edge, we also have the two-sided expansion
 
-  H(binned X)=H(X)-(calH(X)-(dim X)log b+O(n^{-c}).
+  H(binned X)=H(X)-(H(X)-(dim X)log b+O(n^{-c}).
 
 Here is the justification, since everything downstream rests on it.
 Bin averaging changes a Gaussian density, or a mixture of Gaussians, in
@@ -961,20 +961,20 @@ Substituting (eq:ent-edge-concentration) and
 (eq:ent-binning) into (eq:ent-quantized), the mesh terms
 cancel because
 (d+1)ell-frac d2(2ell)=ell, and the shift ε V leaves the
-differential entropy unchanged; since F_{rm global} is a
-deterministic shift of Y_{rm global} we obtain
+differential entropy unchanged; since F_{global} is a
+deterministic shift of Y_{global} we obtain
 (eq:ent-star-edge).
 
 Step 4: the global entropy. The determinant formula for a
 Gaussian gives
 [
-  frac1nH(Y_{rm global})
-  =(ell)/(2)logfrac{2pi{rm e}eta}{pirho}
-   -(ell)/(n)sum_{i=1}^nlog|lambda_i-E-mathrm irmieta|,
+  frac1nH(Y_{global})
+  =(ell)/(2)logfrac{2pi{e}eta}{pirho}
+   -(ell)/(n)sum_{i=1}^nlog|lambda_i-E-irmieta|,
 ]
 because the eigenvalues of Gamma are
 etabigl/(pirho,((lambda_i-E)^2+eta^2)), each with
-multiplicity ell. The function t|->log|t-E-mathrm irmieta| is
+multiplicity ell. The function t|->log|t-E-irmieta| is
 Lipschitz with constant at most eta^{-1}. By rigidity we may replace
 the eigenvalues by Kesten--McKay quantiles at a total error of at most
 n·eta^{-1}· n^{-c_0}; the quantile sum differs from its
@@ -982,12 +982,12 @@ integral against rho_d by O((neta)^{-1}), by summing the
 lengths of the quantile intervals; an omitted or extra bounded
 eigenvalue costs O(log n/n). The choice alpha<c_0 makes the first
 error o(n). Lemma~lem:tree-identities supplies the remaining
-regularization error O(eta) in turning log|t-E-mathrm irmieta| into
+regularization error O(eta) in turning log|t-E-irmieta| into
 log|t-E|. This proves (eq:ent-global-entropy).
 
 
 
-The entropy inequality (eq:ent-star-edge) still contains one
+The entropy ineuality (eq:ent-star-edge) still contains one
 nearly deterministic direction. An eigenvector satisfies
 sum_{ysim v}u(y)=lambda u(v), so at a star with centre v the sum
 of the d neighbour values is nearly E times the centre value; this
@@ -999,17 +999,17 @@ Hermite expansion in the tensor step needs.
 Write a star as (Z,X) with ZinR^{ell} (the ell colour values
 at the centre) and XinR^{d} ⊗ R^{ell} (the colour values at
 the d neighbours). Let P be the orthogonal projection of R^d onto
-{bf 1}^{perp}, the hyperplane orthogonal to the all-ones vector, and
+{ 1}^{perp}, the hyperplane orthogonal to the all-ones vector, and
 define the linear map to the ``clean'' coordinates, and the residual,
 by
 [
-  B(Z,X)=(Z,(PX)/(calB(Z,X)=(Z,(PX)/(sqrt a)),
-  R_{rm res}(Z,X)=sum_{i=1}^dX_i-EZ .
+  B(Z,X)=(Z,(PX)/(B(Z,X)=(Z,(PX)/(sqrt a)),
+  R_{res}(Z,X)=sum_{i=1}^dX_i-EZ .
 ]
 Everything is tensored with the identity on the ell colours, so
-D:=(R ⊕ {bf 1}^{perp}) ⊗ R^{ell} is the space in which
+D:=(R ⊕ { 1}^{perp}) ⊗ R^{ell} is the space in which
 B takes its values. Let w denote the unit vector along the
-central direction R inside R ⊕ {bf 1}^{perp}, and put
+central direction R inside R ⊕ { 1}^{perp}, and put
 [
   v_i=p,w+sqrt a,Pe_i,
   U_i=spanop(w,v_i) ⊗ R^{ell}(i=1,...,d),
@@ -1024,7 +1024,7 @@ they are linearly independent and the coordinate map on U_i is
 invertible.
 
 Let P_i be the orthogonal projection of D onto U_i, put
-B_V=B(V_{rm star}) where V_{rm star} is the true
+B_V=B(V_{star}) where V_{star} is the true
 eigenvector star, and let G be an independent standard Gaussian vector
 in D. All entropies on D and on U_i are computed with their
 Euclidean volume.
@@ -1044,10 +1044,10 @@ interchanges w and v_i and acts trivially on the colours.
 
 Proof. 
 The Jacobian. Describe the map
-(Z,X)|->(B(Z,X),R_calB(Z,X),R_{rm res}(Z,X)) in the orthonormal
-coordinates of R^{d+1}=R ⊕ R{bf 1} ⊕ {bf 1}^{perp}, one
+(Z,X)|->(B(Z,X),R_calB(Z,X),R_{res}(Z,X)) in the orthonormal
+coordinates of R^{d+1}=R ⊕ R{ 1} ⊕ { 1}^{perp}, one
 colour at a time. On the central line it is the identity. On the
-all-ones line, R_{rm res} reads sqrt d,t-EZ where X=t{bf
+all-ones line, R_{res} reads sqrt d,t-EZ where X=t{
 1}/sqrt d+..., so the diagonal entry is sqrt d. On each of the
 d-1 remaining directions, B divides by sqrt a, so the
 diagonal entry is a^{-1/2}. The map is triangular in these
@@ -1065,15 +1065,15 @@ For the eigenvector part, the eigenvector equation gives
 [
   sum_{i=1}^d u(y_i)=lambda,u(v)
   quadsoquad
-  R_{rm res}(V_{rm star})(v)=(lambda-E)V(v),
+  R_{res}(V_{star})(v)=(lambda-E)V(v),
 ]
 and on the graph event |lambda-E|le S^2/n for every selected index.
 Thus, putting Delta=diag(lambda_{i(1)}-E,...,lambda_{i(ell)}-E),
-the deterministic residual of F_{rm star}=ε V_{rm star}+Y_{rm star}
+the deterministic residual of F_{star}=ε V_{star}+Y_{star}
 at a uniform centre has, conditionally on the location, the covariance
 matrix at most
 
-  Cov(R_{rm res}(F_{rm star}))
+  Cov(R_{res}(F_{star}))
   le(eta)/(pirho)Id_{ell}+ε^2Delta^2
   le((eta)/(pirho)+(S^4)/(n^2))Id_{ell},
 
@@ -1082,8 +1082,8 @@ terms vanish, and because |Delta|le S^2/n entrywise. The maximum
 entropy bound by the covariance, together with the fact that
 conditioning decreases entropy, therefore gives
 [
-  H(R_{rm res}(F_{rm star})mid B(F_{rm star}))
-  le(ell)/(2)logfrac{2pi{rm e}eta}{pirho}+O(n^{-c}).
+  H(R_{res}(F_{star})mid B(F_{star}))
+  le(ell)/(2)logfrac{2pi{e}eta}{pirho}+O(n^{-c}).
 ]
 The relative correction inside the logarithm is
 [
@@ -1094,10 +1094,10 @@ which is polynomial because S^4=n^{o(1)} and eta=n^{-alpha} with
 alpha<1.
 
 The noise is standard. By the typical-star estimate of
-Proposition~prop:spectral-input at the height E+mathrm irmieta,
+Proposition~prop:spectral-input at the height E+irmieta,
 together with the tree covariance of Lemma~lem:tree-identities,
 the covariance matrix of the smoothed star noise
-B(Y_{rm star}) differs from Id_D by a polynomial error
+B(Y_{star}) differs from Id_D by a polynomial error
 outside a polynomially small fraction of centres. Two centred Gaussian
 vectors whose covariance matrices differ by delta in operator norm
 have laws at total variation distance at most Cdelta after
@@ -1114,29 +1114,29 @@ absolute determinant (1-p^2)^{ell/2}.
 Bookkeeping: the constants cancel. For a linear bijection Phi
 with Jacobian J_Phi
 [
-  H( H(X)+calH(X)+log|det J_Phi| ,
+  H( H(X)+H(X)+log|det J_Phi| ,
 ]
 since the density of Phi(X) is the density of X divided by
 |det J_Phi|. Applied to the triangular map
-Phi=(Z,X)|->(B(Z,X),R_calB(Z,X),R_{rm res}(Z,X)), whose determinant
+Phi=(Z,X)|->(B(Z,X),R_calB(Z,X),R_{res}(Z,X)), whose determinant
 per colour is sqrt d,a^{-(d-1)/2}, and combining with the
 conditional bound on the residual, we get
 [
-  H(F_{rm star})
-  leH( B(F_calB(F_{rm star}))
-   +(ell)/(2)logfrac{2pi{rm e}eta}{pirho}
+  H(F_{star})
+  leH( B(F_calB(F_{star}))
+   +(ell)/(2)logfrac{2pi{e}eta}{pirho}
    -ell(frac12log d-(d-1)/(2)log a)+O(n^{-c}).
 ]
-The edge observation is the image of P_iF_{rm star} under the
+The edge observation is the image of P_iF_{star} under the
 coordinate map of U_i, which has determinant
 (1-p^2)^{ell/2}, so
 [
-  H(F_{rm edge})
-  =H(P_iF_{rm star})+(ell)/(2)log(1-p^2)+O(n^{-c}).
+  H(F_{edge})
+  =H(P_iF_{star})+(ell)/(2)log(1-p^2)+O(n^{-c}).
 ]
 Substituting both formulas into (eq:ent-star-edge), and using
 (eq:ent-global-entropy) for the right-hand side, the terms with
-log(2pi{rm e}eta/pirho) cancel, and the remaining constant is
+log(2pi{e}eta/pirho) cancel, and the remaining constant is
 [
   -ell U(E)+ell(frac12log d-(d-1)/(2)log a)
   +(dell)/(4)log(1-p^2),
@@ -1144,13 +1144,13 @@ log(2pi{rm e}eta/pirho) cancel, and the remaining constant is
 which is exactly zero by the potential identity
 (eq:tree-potential). Hence
 [
-  H( B(F_calB(F_{rm star}))
-  -frac d2,H(P_iF_{rm star}) ge -O(n^{-c}).
+  H( B(F_calB(F_{star}))
+  -frac d2,H(P_iF_{star}) ge -O(n^{-c}).
 ]
-Replacing the smoothed eigenvector star F_{rm star} by
+Replacing the smoothed eigenvector star F_{star} by
 ε B_V+G costs polynomial entropy in exactly the same way as in
 the noise-replacement step above: the covariance of
-B(F_{rm star}) is Id_D+O(n^{-1+o(1)}) uniformly in
+B(F_{star}) is Id_D+O(n^{-1+o(1)}) uniformly in
 ε, and the entropy is a continuous function of the covariance
 on the fixed finite-dimensional space D. This proves
 (eq:ent-projected-entropy).
@@ -1182,11 +1182,11 @@ acts trivially on the colours. This proves the invariance assertion.
 
 
 
-We have the inequality (eq:ent-projected-entropy), the approximate
+We have the ineuality (eq:ent-projected-entropy), the approximate
 reversal symmetry, and the second moments. What is still missing is the
 inference from a small entropy deficit to small moments.
 Two facts do the work: a formal identity expressing the entropy deficit
-as a difference of relative entropies, and a strict inequality for
+as a difference of relative entropies, and a strict ineuality for
 symmetric tensors that forbids a nonzero first Hermite coefficient.
 
 Write phi_L for the standard Gaussian density on a Euclidean space
@@ -1203,7 +1203,7 @@ Let Z be a centred random vector in D with identity covariance
 matrix and finite differential entropies for Z and for each P_iZ.
 Then
 
-  H(Z)-calH(Z)-frac12sum_{i=1}^dH(P_iZ)
+  H(Z)-H(Z)-frac12sum_{i=1}^dH(P_iZ)
   =-Ent(Law(Z)midphi_D)
    +frac12sum_{i=1}^dEnt(Law(P_iZ)midphi_{U_i}).
 
@@ -1211,13 +1211,13 @@ Then
 
 Proof. 
 For any X in a Euclidean space L with a density,
-H(X)= H(calH(phi_L)-Ent(Law(X)midphi_L); this is
+H(X)= H(H(phi_L)-Ent(Law(X)midphi_L); this is
 the definition of relative entropy together with intphilogphi
-=-H( Also H( L)/(2)log(2pi{rm e}).
+=-H( Also H( L)/(2)log(2pi{e}).
 Substituting into the left side of (eq:ent-deficit-sign), the
 Gaussian entropies contribute
 (dim D)/(2)-frac12sum_i(dim U_i)/(2) times
-log(2pi{rm e}), and this vanishes because
+log(2pi{e}), and this vanishes because
 dim D=dell while dim U_i=2ell and there are d terms. What
 remains is exactly the right side.
 
@@ -1253,8 +1253,8 @@ spanop(w,v_i), which is the same abstract plane for every i since
 w=(1,0) and v_i=(p,√(1-p^2)), define the orthonormal
 bisecting directions
 [
-  e_i^+=frac{w+v_i}{√(2(1+p))},
-  e_i^-=frac{w-v_i}{√(2(1-p))},
+  e_i^+=w+v_i/√(2(1+p)),
+  e_i^-=w-v_i/√(2(1-p)),
   s_i=(e_i^++e_i^-)/(sqrt2),
   t_i=(e_i^+-e_i^-)/(sqrt2).
 ]
@@ -1282,16 +1282,16 @@ particular the two numbers
 are positive and sum to one. With these weights the frame identity
 
   sum_{i=1}^d(b_s,s_is_i^{T}+b_t,t_it_i^{T}
-  =Id_{R ⊕ {bf 1}^{perp}}
+  =Id_{R ⊕ { 1}^{perp}}
 
-holds in R ⊕ {bf 1}^{perp}, tensored with the identity on the
+holds in R ⊕ { 1}^{perp}, tensored with the identity on the
 colours. To verify it, expand both sides in the orthogonal decomposition
-R ⊕ {bf 1} ⊕ {bf 1}^{perp}: the central component of the
+R ⊕ { 1} ⊕ { 1}^{perp}: the central component of the
 left side is
 d(b_sfrac{1+q_0}2+b_tfrac{1-q_0}2)=1; the mixed
-R--{bf 1}^{perp} block vanishes because s_i and t_i have
+R--{ 1}^{perp} block vanishes because s_i and t_i have
 parallel central and Pe_i components with i-independent
-coefficients and sum_iPe_i=0; and the {bf 1}^{perp} block is
+coefficients and sum_iPe_i=0; and the { 1}^{perp} block is
 cP with
 c=b_s(1-q_0)/(2)+b_t(1+q_0)/(2)=1, because
 sum_i(Pe_i)(Pe_i)^{T}
@@ -1328,7 +1328,7 @@ b_s+b_t=1 now gives
 which is the nonnegativity half of (eq:ent-tensor).
 
 Where equality is possible. Suppose equality holds. Then every
-inequality above is an equality, so for every i and every slot
+ineuality above is an equality, so for every i and every slot
 age2 the discarded part vanishes:
 R^{(1)}_{i,s}H=R^{(1)}_{i,s}P_i^{(2)}... P_i^{(j)}H and likewise
 for t. Adding the two identities over the orthonormal pair
@@ -1345,16 +1345,16 @@ Now take ine k and three distinct slots, which exist because jge3:
 Thus the commutator [P_i,P_k] annihilates the first slot of H, for
 all pairs ine k.
 
-Let a_i'in{bf 1}^{perp} be the unit vector along Pe_i. On
-R ⊕ {bf 1}^{perp} the spatial part of this commutator is
+Let a_i'in{ 1}^{perp} be the unit vector along Pe_i. On
+R ⊕ { 1}^{perp} the spatial part of this commutator is
 [
   ip{a_i'}{a_k'}(a_i'(a_k')^{T}-a_k'(a_i')^{T},
    ip{a_i'}{a_k'}=-frac1{d-1}ne0  (dge3),
 ]
 which is an invertible rotation generator on
 spanop(Pe_i,Pe_k) and vanishes on its orthogonal complement. As
-i,k range, these generate all rotations of {bf 1}^{perp}, so
-their common kernel in R ⊕ {bf 1}^{perp} is the central line
+i,k range, these generate all rotations of { 1}^{perp}, so
+their common kernel in R ⊕ { 1}^{perp} is the central line
 R w. Hence H is supported in (w ⊗ R^{ell})^{ ⊗  j}.
 
 But win U_i, so P_i^{ ⊗  j}H=H, and the invariance
@@ -1424,7 +1424,7 @@ proved. Choose
 ]
 and put u=ε/√(1+ε^2). Dividing every entropy
 argument in the hypothesis by √(1+ε^2) leaves that
-inequality unchanged, because the dimensions cancel; the normalized
+ineuality unchanged, because the dimensions cancel; the normalized
 vector is
 [
   Z_u=uB_n+√(1-u^2),G,
@@ -1468,7 +1468,7 @@ values. Consequently
 [
   int_D|f_u-1|^3,ddphi_Dle Cu^3n^{o(1)} ,
 ]
-and, since |(1+h)log(1+h)-h-h^2/2|le C|h|^3 for |h|letfrac12,
+and, since |(1+h)log(1+h)-h-h^2/2|le C|h|^3 for |h|le1/2,
 while int(f_u-1)ddphi_D=0, we get
 
   Ent(Law(Z_u)midphi_D)
@@ -1499,7 +1499,7 @@ O(u^2), because EE|B_n|^2=dim D+o(1). Therefore
 and the same identity holds on U_i with H_j replaced by
 P_i^{ ⊗  j}H_j.
 
-The quadratic-form inequality. Substitute
+The quadratic-form ineuality. Substitute
 (eq:ent-relative-expansion) and its projected analogue into
 (eq:ent-relative-deficit), multiply by 2j!/u^{2j}, and use
 u^{-2j}=n^{2jtau+o(1)}. The result is
@@ -1583,12 +1583,12 @@ the projection step. Then
   =pip{w}{B_V}+ip{Pe_i}{PX}
   =pZ+X_i-frac1dsum_{j=1}^dX_j ,
 ]
-and sum_jX_j=R_{rm res}+EZ by definition of the residual, so
+and sum_jX_j=R_{res}+EZ by definition of the residual, so
 [
   ip{v_i}{B_V}
-  =X_i+(p-frac Ed)Z-frac1dR_{rm res} .
+  =X_i+(p-frac Ed)Z-frac1dR_{res} .
 ]
-Here p=E/d exactly, and |R_{rm res}|=O(n^{-1+o(1)})
+Here p=E/d exactly, and |R_{res}|=O(n^{-1+o(1)})
 because |lambda-E|le S^2/n and |V(v)|=n^{o(1)}. Hence the edge
 value vector equals the (w,v_i)-observations of B_V up to
 O(n^{-1+o(1)}) in every coordinate, and since all coordinates are at
@@ -1687,7 +1687,7 @@ Let W=A-zeta with Imzeta>0, G=W^{-1}, G'=(W+Z)^{-1}, and
   A_{vv}=v^{T},
   D=(1+A_{uv})^2-A_{uu}A_{vv}.
 ]
-Then Dneq0 and
+Then Dne0 and
 
   G'_{xy}
   &=G_{xy}-D^{-1}Bigl[(G_{xv}-G_{xy})
@@ -1767,10 +1767,10 @@ y_{k,D}<∞ with the following property. For every fixed sinR
 and every fixed yge y_{k,D},
 
  limsup_{nto∞}PP(
- |m_n(s+mathrm iy)-m_*|rmiy)-m_*|
- +|M_{n,k}(s+mathrm iy)-K_k(m_*)rmiy)-K_k(m_*)|>y^{-xi})le C_{k,D},y^{-D}.
+ |m_n(s+iy)-m_*|rmiy)-m_*|
+ +|M_{n,k}(s+iy)-K_k(m_*)rmiy)-K_k(m_*)|>y^{-xi})le C_{k,D},y^{-D}.
 
-The constants and the threshold do not depend on s, and s+mathrm iyrmiy
+The constants and the threshold do not depend on s, and s+iyrmiy
 is fixed before the limit.
 
 
@@ -1779,12 +1779,12 @@ It suffices to treat kge2, since smaller matrices are compressions;
 throughout k,s,y are fixed and error exponents may decrease from one
 occurrence to the next.
 
-Initial estimates. At z_0=s+mathrm iS,rmiS,
+Initial estimates. At z_0=s+iS,rmiS,
 Proposition~prop:spectral-input and Lemma~lem:tree-identities
 give, for some c>0,
 
  m_n(z_0)=m_*+O(S^{-c}), M_{n,k}(z_0)=K_k(m_*)+O(S^{-c}),
- Im m_n(mathrm iS^2)+rmiS^2)+|Im M_{n,k}(mathrm iS^2)rmiS^2)|=O(1).
+ Im m_n(iS^2)+rmiS^2)+|Im M_{n,k}(iS^2)rmiS^2)|=O(1).
 
 For the off-block entries, independent uniform vertices v,w satisfy
 EE_{v,w}|R_{vw}|^2=n^{-2}Tr R^*R=Im m_n(z_0)/S, so Markov bounds
@@ -1802,7 +1802,7 @@ zinZ_n,
 
 with c_2>0 independent of D,k,s,C. To see this, compare with z_0:
 for |x|>S^2 and large S,
-|(x-z)^{-1}-(x-z_0)^{-1}|le CSx^{-2}le CS^{-1}Im(x-mathrm iS^2)^rmiS^2)^{-1},
+|(x-z)^{-1}-(x-z_0)^{-1}|le CSx^{-2}le CS^{-1}Im(x-iS^2)^rmiS^2)^{-1},
 so the outside-band parts are O(S^{-1}) (positive residues make the
 matrix bound immediate). Inside the band, with
 w_i(z)=(x_i-z)^{-1}-(x_i-z_0)^{-1}, a bounded trace gives
@@ -1828,22 +1828,22 @@ sampled matching; on that event the switch is a measure-preserving
 involution of the restricted graph/list space. The rank-two update changes the eigenvalue counting
 function by a bounded amount, so
 
- |widehatwidehat m_n(z)-m_n(z)|m_n(z)-m_n(z)|le C/Im z .
+ |mhat_n(z)-m_n(z)|m_n(z)-m_n(z)|le C/Im z .
 
 Write the first two pairs as (v_1,w_1),(v_2,w_2) and set
 f_1=(e_{v_1}-e_{v_2})/sqrt2, f_2=(e_{w_1}-e_{w_2})/sqrt2,
 F=(f_1,f_2), B=0&-2-2&0; the switch adds
 FBF^{T}, and the resolvent identity reads
-(I+UB)widehatwidehat U=UU=U with U=F^{T}-E-z/n)^{-1}F. Assuming
+(I+UB)Uhat=UU=U with U=F^{T}-E-z/n)^{-1}F. Assuming
 |m_n(z)|le C and applying (eq:lim-concentration-implication) and
 (eq:lim-rank-trace) to both data sets,
 
  U=m&qq&m+O((Im z)^{-c_2}),
- widehatwidehat U=m&-qU=m&-q-q&m+O((Im z)^{-c_2}),
+ Uhat=m&-qU=m&-q-q&m+O((Im z)^{-c_2}),
   q=(1+Em)/(d), m=m_n(z),
 
-the sign change in widehatwidehat UU coming from expressing the updated pairs
-in the old columns. The off-diagonal of (I+UB)widehatwidehat U=UU=U gives
+the sign change in UUhat coming from expressing the updated pairs
+in the old columns. The off-diagonal of (I+UB)Uhat=UU=U gives
 
  q^2-q-m^2=O((Im z)^{-c_2}).
 
@@ -1856,10 +1856,10 @@ stays a fixed distance from {Im mge0}. Hence, in the bounded range,
 
 Closing the descent. No trace bound is assumed at lower steps.
 Instead, for eta/2leeta'leeta, term by term
-|x-s-mathrm irmieta'|ge(eta'/eta)|x-s-mathrm irmieta|, so
+|x-s-irmieta'|ge(eta'/eta)|x-s-irmieta|, so
 
- |m_n(s+mathrm irmieta')-m_n(s+mathrm irmieta)|
- le(eta-eta')/(eta')Im m_n(s+mathrm irmieta)leIm m_n(s+mathrm irmieta).
+ |m_n(s+irmieta')-m_n(s+irmieta)|
+ le(eta-eta')/(eta')Im m_n(s+irmieta)leIm m_n(s+irmieta).
 
 Choose a fixed trace range containing the initial value and every
 next-step value of (eq:lim-halving-bound) whenever the preceding
@@ -1889,7 +1889,7 @@ deterministic xi,chi>0 such that the following hold.
 
 For every fixed 0<delta<1, almost surely for all sufficiently large R,
 
- m(e+mathrm iu)=m_*+O(u^rmiu)=m_*+O(u^{-xi}), |e|le R, R^deltale ule R^2,
+ m(e+iu)=m_*+O(u^rmiu)=m_*+O(u^{-xi}), |e|le R, R^deltale ule R^2,
 
 and
 
@@ -1899,8 +1899,8 @@ On a possible extension of the probability space, for every fixed k
 there are standard Gaussian vectors g_jinR^{2k}, independent over
 j conditionally on nu, such that
 
- m(z)=m_{rm r}+PVsum_jfrac1{x_j-z},
- M_k(z)=K_k(m_{rm r})+PVsum_jfrac{C_k^{1/2}g_jg_j^{T}{1/2}}{x_j-z}.
+ m(z)=m_{r}+PVsum_jfrac1{x_j-z},
+ M_k(z)=K_k(m_{r})+PVsum_jfrac{C_k^{1/2}g_jg_j^{T}{1/2}}{x_j-z}.
 
 These representations are consistent as k grows by keeping the first
 k channels, a separate mark is assigned to each occurrence of a
@@ -1911,13 +1911,13 @@ for all finite k and all zinC_+.
 
 Proof. 
 Tightness. For fixed L,y>0, positivity gives
-nu_n([-L,L])le(L^2+y^2)y^{-1}Im m_n(mathrm iy);rmiy); using
+nu_n([-L,L])le(L^2+y^2)y^{-1}Im m_n(iy);rmiy); using
 Proposition~prop:endpoint-control with large fixed y (chosen per
 tolerance) gives tightness of the local counts. For compact-open
 tightness, fix a compact K+ and y>0. For a resolvent
 sum f with nonnegative residues, comparing kernels on the real line
-gives Im f(z)le C_{K,y}calK,yIm f(mathrm iy)rmiy) and
-|f(z)-f(mathrm iy)|rmiy)|le C_{K,y}calK,yIm f(mathrm iy)rmiy) for
+gives Im f(z)le C_{K,y}K,yIm f(iy)rmiy) and
+|f(z)-f(iy)|rmiy)|le C_{K,y}K,yIm f(iy)rmiy) for
 zinK; applying these to m_n and, by polarization, to the
 real quadratic forms of M_{n,k} bounds each fixed compact, and Cauchy's
 estimates give equicontinuity on smaller compacts. Extracting diagonally
@@ -1949,18 +1949,18 @@ b_j=C_k^{1/2}g_j to each occurrence of a repeated point.
 
 Removing imaginary tails. For fixed zinC_+ and large L, the
 scalar kernels satisfy, for |x|>L,
-Im(x-z)^{-1}+|(x-z)^{-1}-(x-mathrm i)^rmi)^{-1}|le C_zL^{-1}Im(x-mathrm iL)^rmiL)^{-1};
+Im(x-z)^{-1}+|(x-z)^{-1}-(x-i)^rmi)^{-1}|le C_zL^{-1}Im(x-iL)^rmiL)^{-1};
 for matrix sums use positivity of the residues and bound the norm by the
 trace of the positive imaginary part. By Proposition~prop:endpoint-control
-these traces at mathrm iLrmiL are tight as Lto∞ after the upper
+these traces at iLrmiL are tight as Lto∞ after the upper
 limit in n; the factor L^{-1} removes the tails in probability, and
 compact marked sums converge, so
 
  
  Im m(z)&=sum_jImfrac1{x_j-z},&
- m(z)-m(mathrm i)&=rmi)&=sum_j(frac1{x_j-z}-frac1{x_j-mathrm i}rmi),
+ m(z)-m(i)&=rmi)&=sum_j(frac1{x_j-z}-frac1{x_j-i}rmi),
  Im M_k(z)&=sum_jb_jb_j^{T}{x_j-z},&
- M_k(z)-M_k(mathrm i)&=rmi)&=sum_jb_jb_j^{T}{x_j-z}-frac1{x_j-mathrm i}rmi).
+ M_k(z)-M_k(i)&=rmi)&=sum_jb_jb_j^{T}{x_j-z}-frac1{x_j-i}rmi).
  
 
 No imaginary spectral mass is lost at infinity; real additive constants
@@ -1968,9 +1968,9 @@ are not yet excluded and are fixed below.
 
 Uniform bounds after the limit. Pass
 Proposition~prop:endpoint-control to the limit by the portmanteau
-inequality on the open event of large error (its constants are independent
+ineuality on the open event of large error (its constants are independent
 of s), and slightly decrease xi<1. Fix 0<delta<1 and take a unit
-grid on the padded region |e|le2R, tfrac12R^deltale ule2R^2,
+grid on the padded region |e|le2R, 1/2R^deltale ule2R^2,
 giving O(R^3) arguments. For fixed dyadic R this is a finite
 collection in the exact limit; the failure probability is at most
 CR^{3-delta D}, and Borel--Cantelli over dyadic R with large D
@@ -1981,11 +1981,11 @@ countable set of delta's gives every fixed-delta conclusion. The
 limit in n preceded this growing-grid argument.
 
 From smoothing to counts. Equation~(eq:mesoscopic-limit) gives
-nu([-R,R])=O(R). Integrating pi^{-1}Im m(e+mathrm irmisqrt R) over
+nu([-R,R])=O(R). Integrating pi^{-1}Im m(e+irmisqrt R) over
 ein[v,w] yields rho(w-v)+O(R^{1-xi/2}). Compare with the sharp
 count using P_{v,w}(x)=pi^{-1}int_v^w(sqrt R)/((x-e)^2+R)dd e:
 points within R^{3/4} of an endpoint number O(R^{3/4}); on
-[-2R,2R] off the two strips, |P_{v,w}(x)-mathbf 1_bf1_{[v,w]}(x)|=O(R^{-1/4}),
+[-2R,2R] off the two strips, |P_{v,w}(x)-1_bf1_{[v,w]}(x)|=O(R^{-1/4}),
 for total error O(R^{3/4}); for |x|>2R the bound
 P_{v,w}le CR^{3/2}/x^2 with the linear count contributes
 O(sqrt R). This gives (eq:discrepancy) for any
@@ -1997,25 +1997,25 @@ sum_{|x_j|>R}x_j^{-2}=O(R^{-1}) (the leading density terms cancel in
 the symmetric sum on the two half-lines). Hence
 S_nu(z)=PVsum_j(x_j-z)^{-1} is analytic and locally uniformly
 convergent, and comparing with the constant density gives
-S_nu(mathrm iR)=rmiR)=mathrm irmipirho+o(1). By
+S_nu(iR)=rmiR)=irmipirho+o(1). By
 (eq:lim-difference-identities), m-S_nu is constant with zero
-imaginary part; the two displays identify it as m_{rm r}.
+imaginary part; the two displays identify it as m_{r}.
 
 Marked principal values. Fix k, condition on nu, and set
 Z_j=b_jb_j^{T}-C_k, independent centered with bounded second
-moments. Ordered by increasing |x_j|, sum_jZ_j/(x_j-mathrm i)rmi)
+moments. Ordered by increasing |x_j|, sum_jZ_j/(x_j-i)rmi)
 converges almost surely entrywise, and
 sum_{|x_j|>1}|Z_j|x_j^{-2}<∞ conditionally almost surely, so
-the difference series between z and mathrm irmi converges absolutely
+the difference series between z and irmi converges absolutely
 and locally uniformly; with the scalar PV this gives local uniform
 convergence of S_{k,nu}(z)=PVsum_jb_jb_j^{T}-z)^{-1}.
-The linear count gives sum_j|x_j-mathrm iR|^rmiR|^{-2}=O(R^{-1}), so each
-entry of the centered marked series at mathrm iRrmiR has conditional
+The linear count gives sum_j|x_j-iR|^rmiR|^{-2}=O(R^{-1}), so each
+entry of the centered marked series at iRrmiR has conditional
 second moment O(R^{-1}), whence
-S_{k,nu}(mathrm iR)rmiR)tomathrm irmipirho,C_k in probability. Since
+S_{k,nu}(iR)rmiR)toirmipirho,C_k in probability. Since
 M_k-S_{k,nu} is almost surely constant in z and
-M_k(mathrm iR)rmiR)to K_k(m_*) in probability, that constant equals
-K_k(m_*)-mathrm irmipirho,C_k=K_k(m_{rm r}).
+M_k(iR)rmiR)to K_k(m_*) in probability, that constant equals
+K_k(m_*)-irmipirho,C_k=K_k(m_{r}).
 
 Finally, (eq:discrepancy) supplies infinitely many points and C_k
 is positive definite; 2k independent Gaussian marks span R^{2k}
@@ -2037,7 +2037,7 @@ its Gaussian endpoint marks. Those marks give a symmetry that a general
 law would not have: conjugating M_k by an orthogonal Q_U preserves
 the law. Composing many small conjugations then produces an honest GOE
 perturbation: the natural graph trace S_r(z) is approximated,
-in law and up to o(1), by m_{rm r} + Tr(H_r-z)^{-1}, where H_r
+in law and up to o(1), by m_{r} + Tr(H_r-z)^{-1}, where H_r
 is a fixed diagonal plus independent GOE noise.
 
 
@@ -2068,20 +2068,20 @@ W|-> WD_O permutes the pairing between the two endpoint lists, is a
 bijection of the restricted graph/list space preserving all weights, and
 is inverted by the inverse permutation. The determinant and inverse
 identities for finite-rank perturbations give
-widehatwidehat m_n=m_n-∂_zm_n=m_n-∂_zlogdet(I_{2k}+B_OM_{n,k}) and
-widehatwidehat M_M_{n,k}=D_O^{T}{n,k}^{-1}+B_O)^{-1}D_O (the factor
+mhat_n=m_n-∂_zm_n=m_n-∂_zlogdet(I_{2k}+B_OM_{n,k}) and
+Mhat_M_{n,k}=D_O^{T}{n,k}^{-1}+B_O)^{-1}D_O (the factor
 n^{-1} in m_n=n^{-1}Tr R supplies the derivative). Compact-open
 convergence of analytic functions gives convergence of derivatives, and
 positive-definite imaginary parts make the inverse regular, so passing to
 the limit proves the claim for permutation matrices.
 
-For Uin O(k) put Q_U=diag(U,U). As C_k and K_k(m_{rm r})
+For Uin O(k) put Q_U=diag(U,U). As C_k and K_k(m_{r})
 commute with Q_U, the Gaussian representation
 (eq:marked-representation) shows that
-R_U(m,M)=(m,Q_U^calR_U(m,M)=(m,Q_U^{T} preserves the joint law. Since
+R_U(m,M)=(m,Q_U^R_U(m,M)=(m,Q_U^{T} preserves the joint law. Since
 Q_UD_OQ_U^{T}{UOU^{T}} and
 Q_UB_OQ_U^{T}{UOU^{T}}, we get
-R_U^{-1}T_O R_U=calR_U=T_{UOU^{T}};
+R_U^{-1}T_O R_U=R_U=T_{UOU^{T}};
 so the law is invariant under every orthogonal conjugate of a permutation
 switch. The maps satisfy B_{OO'}=B_O+D_OB_{O'}D_O^{T} and
 D_{OO'}=D_OD_{O'}, so T_O T_calT_{O'}=T_{OO'}
@@ -2102,7 +2102,7 @@ the principal-value resolvent into a finite matrix. Choose once and for all
  theta=√(T/r), L=r^{100/chi}, rinN .
 
 For each r let O=O_r be the direct sum of r planar rotations of angle
-theta, write B=B_{O_r} and M^0=K_k(m_{rm r}). The real symmetric
+theta, write B=B_{O_r} and M^0=K_k(m_{r}). The real symmetric
 matrix
 
  Gamma_r=C_k^{1/2}(B^{-1}+M^0)^{-1}C_k^{1/2}
@@ -2129,7 +2129,7 @@ The coefficients satisfy
 
 with deterministic positive constants. As rto∞,
 
- m_{rm r}+Tr(H_r-z)^{-1} ⇒ m(z)
+ m_{r}+Tr(H_r-z)^{-1} ⇒ m(z)
 
 jointly at every fixed finite list of zinC_+, and the unscaled
 eigenvalue point measure of H_r converges locally vaguely in law to
@@ -2138,7 +2138,7 @@ conditionally on nu separately for each r, and writing
 S_r(z)=m(z)-∂_zcalS_r(z)=m(z)-∂_zlogdet(I_{2k}+BM_k(z)), expectations of
 every bounded smooth function with bounded derivatives of finitely many
 real and imaginary trace values differ by o(1) when
-m_{rm r}+Tr(H_r-z)^{-1} is replaced by S_r(z).
+m_{r}+Tr(H_r-z)^{-1} is replaced by S_r(z).
 
 
 First cut at L. Elementary consequences of (eq:discrepancy)
@@ -2167,7 +2167,7 @@ gives
 with probability tending to one (using Lgg r^4). As B^{-1} is real
 symmetric, this bounds |(B^{-1}+M)^{-1}| and |(B^{-1}+M_L)^{-1}| by
 r^5, so cutting both scalar and marked sums in
-S_r(z)=m(z)-calS_r(z)=m(z)-Tr(B^{-1}+M(z))^{-1}M'(z) costs o_{PP}(1): the
+S_r(z)=m(z)-S_r(z)=m(z)-Tr(B^{-1}+M(z))^{-1}M'(z) costs o_{PP}(1): the
 marked error is O_{PP}(r· r^5· r^{-98})+O_{PP}(r· r^{10}
 · r^{-98}· r^2)=O_{PP}(r^{-85}), and the scalar tail is
 O(L^{-chi}). Diagonalizing Gamma_r and rotating the independent
@@ -2175,8 +2175,8 @@ Gaussian row vectors in channel space gives a matrix G of
 independent standard Gaussian entries with 4r columns, and the
 finite-rank inverse identity (Woodbury) yields
 
- S_r(z)=m_{rm r}+Tr(D+A'-z)^{-1}+o_{PP}(1),
- A'=G G^calG^top, D=diag(x_j:|x_j|le L).
+ S_r(z)=m_{r}+Tr(D+A'-z)^{-1}+o_{PP}(1),
+ A'=G G^G^top, D=diag(x_j:|x_j|le L).
 
 The coefficient bounds follow from
 (B^{-1}+M^0)^{-1}=(Id+BM^0)^{-1}B=B+O(theta^2): every singular value
@@ -2198,7 +2198,7 @@ of log r whose exponent may rise from one occurrence to the next.
 For A{I,F} put a_I=h, a_F=K. With probability tending
 to one,
 
- bigl|U_{A}'_{A A}calA-s_1Id)U_{A}|
+ bigl|U_{A}'_{A A}A-s_1Id)U_{A}|
  lepolylog(r)(sqrt{T/a_{A}}+theta),
 
 and, for P_0=U_IA'_{IF}U_F,
@@ -2218,8 +2218,8 @@ superpolynomially small centering correction) leaves summands of norm at
 most C|beta|(log r)^2Tr Plethetapolylog(r), and
 bounds the matrix variance parameter by Cs_2(Tr P)|P|le
 (T/a_{A} The bounded matrix Bernstein
-inequality gives a two-sided tail 2|A|calA|exp(-ct^2/
-(sigma_{rm mat}^2+bt)), and a large power of log r in t
+ineuality gives a two-sided tail 2|A|A|exp(-ct^2/
+(sigma_{mat}^2+bt)), and a large power of log r in t
 proves (eq:sw-bernstein). For the cross block, independence of
 distinct Gaussian rows gives EE|P_0|_{HS}^2=s_2(Tr U_I^2)(Tr U_F^2),
 so Markov with (eq:sw-weight-bounds) gives (eq:sw-cross-block).
@@ -2253,7 +2253,7 @@ Q_F=U_F^{-1}G_FU_F^{-1} has bounded norm. Also
 For Σ, replacing G_F by D_F^{-1} costs at most
 |P_0|_{HS}^2delta_Fle Tpolylog(r)delta_F in the
 U_I-weighted norm. For A'_{IF}D_F^{-1}A'_{FI}, condition on the inner
-block and set Q=G_I G_I^calG_I^top; the columns a_j
+block and set Q=G_I G_I^G_I^top; the columns a_j
 of A'_{IF}, jin F, are independent centered Gaussian with covariance
 Q, so EE[sum_{jin F}a_ja_j^top/x_jmid·]=Qsum_{jin F}x_j^{-1}
 and the entrywise variance is (Q_{aa}Q_{bb}+Q_{ab}^2)sum_{jin F}x_j^{-2}.
@@ -2263,8 +2263,8 @@ EE|U_IA'_{IF}D_F^{-1}A'_{FI}U_I|_{HS}^2le C T^2(log r)^2
 (K^{-2chi}+K^{-1}), so by Markov this norm is
 Tpolylog(r)(K^{-chi}+K^{-1/2}) with probability tending to
 one. Combining and multiplying by T^5, the exponents
-6tau-chi/100, tfrac{13}2tau-tfrac1{200} and
-tfrac{13}2tau-tfrac12 are all negative under (eq:sw-parameters),
+6tau-chi/100, tfrac{13}2tau-1/{200} and
+tfrac{13}2tau-1/2 are all negative under (eq:sw-parameters),
 giving (eq:sw-schur-small).
 
 
@@ -2281,16 +2281,16 @@ jointly for any fixed finite list of z's.
 
 
 Proof. 
-Every diagonal entry of U_I(D_I-mathrm ih)U_Irmih)U_I has modulus at least
+Every diagonal entry of U_I(D_I-ih)U_Irmih)U_I has modulus at least
 1/sqrt2, and the added term has norm o(1) by
 Lemma~sw:weighted-blocks, |s_1|=O(T) and (eq:sw-weight-bounds).
-Hence |U_I^{-1}G_I(mathrm ih)U_I^rmih)U_I^{-1}|le C, and
-G_I(mathrm ih)^*G_I(rmih)^*G_I(mathrm ih)=h^rmih)=h^{-1}Im G_I(mathrm ih)rmih) gives
-|G_I(mathrm ih)U_I^rmih)U_I^{-1}|le Ch^{-1/2}. The twice-iterated resolvent
-identity G_I(z)=G_I(mathrm ih)+(z-rmih)+(z-mathrm ih)G_I(rmih)G_I(mathrm ih)^2+rmih)^2+
-(z-mathrm ih)^2G_I(rmih)^2G_I(mathrm ih)G_I(z)G_I(rmih)G_I(z)G_I(mathrm ih),rmih), conjugated by
-U_I^{-1}, bounds the three terms by C, C|z-mathrm ih|/hrmih|/h and
-C|z-mathrm ih|^2/(hrmih|^2/(hIm z), proving (eq:sw-inner-resolvent).
+Hence |U_I^{-1}G_I(ih)U_I^rmih)U_I^{-1}|le C, and
+G_I(ih)^*G_I(rmih)^*G_I(ih)=h^rmih)=h^{-1}Im G_I(ih)rmih) gives
+|G_I(ih)U_I^rmih)U_I^{-1}|le Ch^{-1/2}. The twice-iterated resolvent
+identity G_I(z)=G_I(ih)+(z-rmih)+(z-ih)G_I(rmih)G_I(ih)^2+rmih)^2+
+(z-ih)^2G_I(rmih)^2G_I(ih)G_I(z)G_I(rmih)G_I(z)G_I(ih),rmih), conjugated by
+U_I^{-1}, bounds the three terms by C, C|z-ih|/hrmih|/h and
+C|z-ih|^2/(hrmih|^2/(hIm z), proving (eq:sw-inner-resolvent).
 
 For (eq:sw-second-cut), the upper-left block of the full inverse is
 widetilde G_I=(G_I^{-1}-Σ)^{-1}. With R_I=U_I^{-1}G_IU_I^{-1} and
@@ -2343,9 +2343,9 @@ this proves (eq:sw-trace-limit).
 
 For the point measures, let nu^{(r)} be the eigenvalue counting measure
 of H_r and sigma_r(dd x)=nu^{(r)}(dd x)/(1+x^2); its total mass
-ImTr(H_r-mathrm i)^rmi)^{-1} is tight by (eq:sw-trace-limit), so
+ImTr(H_r-i)^rmi)^{-1} is tight by (eq:sw-trace-limit), so
 sigma_r is tight on the one-point compactification overlineR. For
-z=e+mathrm irmieta the kernel k_z(x)=(1+x^2)eta((x-e)^2+eta^2)^{-1}
+z=e+irmieta the kernel k_z(x)=(1+x^2)eta((x-e)^2+eta^2)^{-1}
 extends continuously to overlineR with k_z(∞)=eta, and its
 integral against sigma_r is ImTr(H_r-z)^{-1}; so any subsequential
 limit sigma has the law of Im m at a countable dense set of z,
@@ -2380,7 +2380,7 @@ For a real diagonal V=diag(v_1,...,v_m) write
 m_V(zeta)=m^{-1}sum_j(v_j-zeta)^{-1}. For g,G>0 call V
 (g,G)-regular if, for fixed c,C,C_V>0,
 
- cleIm m_V(e+mathrm irmieta)le Cquad (|e|le G, gleetale10),
+ cleIm m_V(e+irmieta)le Cquad (|e|le G, gleetale10),
   |V|le m^{C_V},
 
 with g,G possibly m-dependent and, for a fixed delta>0,
@@ -2391,8 +2391,8 @@ Let W_m be standard normalized GOE (off-diagonal variance 1/m,
 diagonal 2/m). The free-convolution comparison density associated with
 V+sqrt t,W_m is described by
 
- m_{{rm fc},t}(zeta)=m_V(zeta+t,m_{{rm fc},t}(zeta)),
- rho_{{rm fc},t}(e)=pi^{-1}lim_{eta↓0}Im m_{{rm fc},t}(e+mathrm irmieta),
+ m_{{fc},t}(zeta)=m_V(zeta+t,m_{{fc},t}(zeta)),
+ rho_{{fc},t}(e)=pi^{-1}lim_{eta↓0}Im m_{{fc},t}(e+irmieta),
 
 using the upper-half-plane solution. Write p_H^{(m)} for the unit-mass
 symmetrized joint eigenvalue density of H, and p_H^{(a)} for its
@@ -2405,7 +2405,7 @@ with fixed constants. Fix sigma>0 and
 
  g,m^{sigma}le tle m^{-sigma}G^2,
 
-and assume rho_{{rm fc},t}(0)=1/pi. Then for every fixed age1 and
+and assume rho_{{fc},t}(0)=1/pi. Then for every fixed age1 and
 Oin C_c^∞(R^a) the difference of
 int O(boldsymbolalpha),p^{(a)}_{V+sqrt tW_m}(piboldsymbolalpha/m)ddboldsymbolalpha
 and int O(boldsymbolalpha),p^{(a)}_{W_m}(piboldsymbolalpha/m)ddboldsymbolalpha
@@ -2446,7 +2446,7 @@ For almost every nu, V_* is eventually regular with
 g=T^{1/2}/q_*, G=q_*^{-1/8}, satisfying
 (eq:dbm-parameter-domain) and (eq:dbm-time-window) with
 m=q_*, delta=sigma=tau. Moreover
-rho_{{rm fc},t_*}(0)to1/pi; with c_r=pirho_{{rm fc},t_*}(0) the
+rho_{{fc},t_*}(0)to1/pi; with c_r=pirho_{{fc},t_*}(0) the
 data c_rV_*, c_r^2t_* satisfy Theorem~thm:dbm-input, including
 the exact density condition.
 
@@ -2475,13 +2475,13 @@ For the density, zeta=0 in (eq:dbm-free-convolution) reads
 
  w=F_r(w), F_r(w)=alpha_*^{-1}sum_{jin I}(x_j-(alpha_*s_2w-s_1))^{-1}.
 
-On |w-mathrm i|rmi|le1/4 the argument z=alpha_*s_2w-s_1 has
+On |w-i|rmi|le1/4 the argument z=alpha_*s_2w-s_1 has
 Im zasymp T, |z|=O(T); (eq:mesoscopic-limit) and
 (eq:marked-representation) give
-PVsum_j(x_j-z)^{-1}=mathrm irmipirho+o(1), and
-PVsum_{|x_j|>K}(x_j-z)^{-1}=O(K^{-chi}+T/K). Hence F_r(w)=mathrm i+o(1)rmi+o(1)
-uniformly; Rouch'e for w-F_r(w) against w-mathrm irmi gives a unique
-simple zero w_rtomathrm irmi in the disk, continued by the implicit
+PVsum_j(x_j-z)^{-1}=irmipirho+o(1), and
+PVsum_{|x_j|>K}(x_j-z)^{-1}=O(K^{-chi}+T/K). Hence F_r(w)=i+o(1)rmi+o(1)
+uniformly; Rouch'e for w-F_r(w) against w-irmi gives a unique
+simple zero w_rtoirmi in the disk, continued by the implicit
 function theorem to a positive-imaginary solution near zeta=0. That this
 is the physical branch follows from the identity
 
@@ -2492,12 +2492,12 @@ for every upper-half-plane solution; two distinct solutions would give,
 after subtracting and dividing, 1 equal to a quantity of absolute value
 strictly less than one by Cauchy--Schwarz and
 (eq:dbm-uniqueness-identity). Its boundary value gives
-rho_{{rm fc},t_*}(0)=Im w_r/pito1/pi.
+rho_{{fc},t_*}(0)=Im w_r/pito1/pi.
 
 Finally, replacing V_*,t_* by c_rV_*,c_r^2t_* dilates the
 free-convolution law by c_r, with transform
-c_r^{-1}m_{{rm fc},t_*}(zeta/c_r) and density
-c_r^{-1}rho_{{rm fc},t_*}(0)=1/pi at zero; since c_rto1 the enlarged
+c_r^{-1}m_{{fc},t_*}(zeta/c_r) and density
+c_r^{-1}rho_{{fc},t_*}(0)=1/pi at zero; since c_rto1 the enlarged
 domain gives (eq:dbm-regularity) and the strict power margins
 preserve (eq:dbm-time-window) for c_r^2t_*.
 
@@ -2520,17 +2520,17 @@ G_m.
 
 
 For every compact JsubsetR and every b>0,
-sup_{mge1}EEexp(b,G_m(J))<∞.calG_m(J))<∞.
+sup_{mge1}EEexp(b,G_m(J))<∞.G_m(J))<∞.
 
 
 Proof. 
 Bound the GOE count by twice a Gaussian-unitary count plus one via the
-Gaussian decimation identity of Forrester--Rains. Let mathrm O_mrmO_m be the
+Gaussian decimation identity of Forrester--Rains. Let O_mrmO_m be the
 decreasingly ordered points with density
 proptoDelta(y)prod_i b_0(y_i), b_0=e^{-y^2/2}, the law of
-sqrt m,W_m/sqrt2; then G_m(J) is the mathrm O_mrmO_m count in
+sqrt m,W_m/sqrt2; then G_m(J) is the O_mrmO_m count in
 J_m=pi J/√(2m), so |J_m|=O(m^{-1/2}). Superposing independent
-mathrm O_mrmO_m and mathrm O_rmO_{m+1} gives density
+O_mrmO_m and O_rmO_{m+1} gives density
 P(z)=sum_{|S|=m}Delta(z_S)Delta(z_{S^c}), which by the
 parity--Vandermonde factorization equals
 
@@ -2557,7 +2557,7 @@ K_m(x,x)le2√(2m). Then for every age0,
 EE(N_m^{even})_ale(2√(2m),|J_m|)^ale C_J^a, so
 EE e^{vN_m^{even}}leexp(C_J(e^v-1)). As the
 superposition count is at most twice its even count plus one and dominates
-the mathrm O_mrmO_m count,
+the O_mrmO_m count,
 EE e^{bG_m(J)} e^bEE e^{2bN_m^{even}}.
 
 
@@ -2570,7 +2570,7 @@ Proposition~prop:gaussian-insertion,
 unconditionally in local vague topology. Fix 0le hin C_c^∞(R),
 f=1-e^{-h}, and for a finite point measure zeta=sum_idelta_{y_i} set
 A_0=1, A_a=(a!)^{-1}sum_{i_1,...,i_a distinct}prod_{j=1}^a f(y_{i_j}),
-B_ell=sum_{a=0}^{ell}(-1)^aA_a. The Bonferroni inequalities give
+B_ell=sum_{a=0}^{ell}(-1)^aA_a. The Bonferroni ineualities give
 
  B_{2ell+1}(zeta)leexp(-int h,ddzeta)le B_{2ell}(zeta),
 
@@ -2583,7 +2583,7 @@ the brackets uniformly in q_*, take Jsupsetsupp h; since
 A_a(G_m){G_m(J)}{-a}3^{G_m(J)}
 (from sum_a2^abinom Na=3^N), Lemma~lem:goe-counts gives
 
- sup_mEE[B_{2ell}(G_m)-B_calG_m)-B_{2ell+1}(G_m)]calG_m)]le C_J2^{-(2ell+1)}.
+ sup_mEE[B_{2ell}(G_m)-B_calG_m)-B_{2ell+1}(G_m)]G_m)]le C_J2^{-(2ell+1)}.
 
 Using (eq:dbm-bonferroni) for both models, sending rto∞ at
 fixed ell and then ellto∞, gives