diff --git a/arithmeticity_local_field_lattices_v2.txt b/arithmeticity_local_field_lattices_v2.txt
index a3f71b0..671d506 100644
--- a/arithmeticity_local_field_lattices_v2.txt
+++ b/arithmeticity_local_field_lattices_v2.txt
@@ -107,7 +107,7 @@ group, like the integer points
 ]
 inside the group of all translations, or the famous
 [
- SL_n(Z) R)bbR)
+ SL_n(Z) R)R)
 ]
 of integer matrices of determinant one inside all real matrices of
 determinant one. A lattice is discrete, so its elements do not
@@ -228,7 +228,7 @@ relation.
 
 The central question becomes: where can finite correspondences come
 from? One family of examples is always available. Suppose Gamma and
-Λ are groups and AleqGamma, BleqΛ are subgroups of
+Λ are groups and AleGamma, BleΛ are subgroups of
 finite index, with an actual isomorphism delta:Ato B between them.
 Suppose moreover there is a finite-dimensional projective unitary
 representation sigma of A whose multiplier matches the cocycle
@@ -253,8 +253,8 @@ let Λ be any countably infinite ICC group, and let mu,omega
 be normalized scalar cocycles. Every bifinite
 L_mu(Gamma)--L_omega(Λ) correspondence is unitarily
 isomorphic to a closed bimodule summand of a finite direct sum of
-arithmetic models E(A,B,delta,sigma), where AleqGamma and
-BleqΛ have finite index, delta:Ato B is an actual group
+arithmetic models E(A,B,delta,sigma), where AleGamma and
+BleΛ have finite index, delta:Ato B is an actual group
 isomorphism, and sigma is a finite-dimensional projective unitary
 representation satisfying the matched-multiplier identity above.
 
@@ -351,7 +351,7 @@ dimensions over the prime field and the upper bound ell_T(g)le
 d_St+O(1).
  Non-isolated archimedean place (Section~sec:panels):
 optimal finite-dimensional modules of the rank-one Levi factor, and
-L^2-trace inequalities on compact operators reconstructed from the
+L^2-trace ineualities on compact operators reconstructed from the
 graph relation, give the analogous bound.
  Isolated real place (Sections~sec:ray--sec:scale):
 ray densities on the symmetric space, a uniform radial-multiplier
@@ -380,7 +380,7 @@ different labels must commute. This is not automatic; they are
 a priori noncommuting operators. Section~sec:locality proves
 commutation for all labels of an isolated factor by a deep
 finite-dimensional locality argument: it builds a graph domain
-mathfrak D,frakD, proves that the coefficients of an arbitrary connected
+D,D, proves that the coefficients of an arbitrary connected
 piece are finite-jets, uses the geometry of the boundary kernel
 K(w,z) to eliminate positive differential order, and concludes that
 the joint spectral law of the angle-height operators is exactly the law
@@ -438,10 +438,10 @@ group is available on the comparison side at the start.
 Let K be the class of countably infinite ICC groups abstractly
 commensurable with a lattice in a finite nonempty product
 [
- G=prod_{i=1}^r G_i, G_i=mathbf H_i(k_i)^+.bfH_i(k_i)^+.
+ G=prod_{i=1}^r G_i, G_i=H_i(k_i)^+.H_i(k_i)^+.
 ]
 Here k_i is R, C, or a finite extension of
-Q_p; the primes may differ. The algebraic group mathbf H_ibfH_i is
+Q_p; the primes may differ. The algebraic group H_i is
 connected, adjoint, and absolutely simple over k_i. The superscript +
 denotes the subgroup generated by the k_i-points of the unipotent
 radicals of proper k_i-parabolic subgroups. Each G_i is required to be
@@ -469,7 +469,7 @@ All scalar cocycles are T-valuedbbT-valued and normalized:
  mu(e,g)=mu(g,e)=1.
 ]
 The twisted group factor L_mu(Gamma) has canonical unitaries u_g
-with u_gu_h=mu(g,h)u_{gh} and trace tau(u_g)=mathbf 1_bf1_{{e}}(g).
+with u_gu_h=mu(g,h)u_{gh} and trace tau(u_g)=1_bf1_{{e}}(g).
 For an ICC group this is a factor: the absolute values of Fourier
 coefficients of a central vector are constant on conjugacy classes, so
 square summability kills all nonidentity coefficients.
@@ -586,7 +586,7 @@ the lattice geometry developed here.
 
 Finite correspondences retain finite-index relations and their
 compositions. The framework is supplied by Jones' index and module
-dimensions [cite], the finite-index expectation inequality of
+dimensions [cite], the finite-index expectation ineuality of
 Pimsner--Popa [cite], and Connes' relative tensor product
 cite[Chapter~V, Appendix~B]{Connes1994}. Vaes' classification for
 generalized Bernoulli factors already organizes finite-index bimodules by
@@ -693,7 +693,7 @@ logarithm argument recover homogeneous residue maps together with upper
 speed bounds (Section~sec:finite).
 
 Real panels inside higher-rank components use optimal finite-dimensional
-modules: trace inequalities force a nonzero scalar witness in their
+modules: trace ineualities force a nonzero scalar witness in their
 section spaces, and positivity of its algebraic action excludes excess
 speed (Section~sec:panels).
 
@@ -869,7 +869,7 @@ gamma(a,delta(a))=sigma(a). Then
 ]
 The unitary sends the vector xi in the fibre labelled x to
 U_{r_x}jxi, where
-jxi=widehatwidehat11 ⊗ (xi ⊗ widehatwidehat1)1) in the displayed model.
+jxi=hathat11 ⊗ (xi ⊗ hathat1)1) in the displayed model.
 
 
 Proof. 
@@ -1048,7 +1048,7 @@ The spaces V_c are finite-dimensional. The v^c_h are normalized
 projective matrix representations with multipliers nu_c, and
 
  rho'_c(h)rho'_c(k)
- =frac{omega(h,k)bar}{nu_c(h,k)}rho'_c(hk).
+ =omega(h,k)bar/nu_c(h,k)rho'_c(hk).
  
 
 Each K'_c is a finite normal module on both sides and satisfies
@@ -1087,7 +1087,7 @@ summable, and every nonidentity class is infinite, so only the identity
 vector remains. The trace-preserving expectation
 E_{N^{op}}:Pto N^{op} is equivariant and
 L^2-contractive, hence E_{N^{op}}(x)=tr(x)1
-for xin D_c. By the finite-index inequality there is ε>0
+for xin D_c. By the finite-index ineuality there is ε>0
 with E_{N^{op}}(x)geε x for xge0, so a nonzero
 projection qin D_c has tr(q)geε. A finite
 algebra with this lower bound has only finitely many orthogonal nonzero
@@ -1250,8 +1250,8 @@ w_r. Thus W_r is right multiplication by w_r^*, which has
 multiplier alpha_R. Set
 
  Y=G/Gamma,
- H=L^2(Y;H),calH=L^2(Y;H),
- N=L^∞(Y)calN=L^∞(Y)overlineoverline ⊗⊗  eM_k(B)e.
+ H=L^2(Y;H),H=L^2(Y;H),
+ N=L^∞(Y)N=L^∞(Y)overlineoverline ⊗⊗  eM_k(B)e.
  
 
 Give Y its invariant probability measure. The trace on N is
@@ -1286,7 +1286,7 @@ W|_{R'_j}. Then
  (A_j)^{beta_j}=N,
  (A_j)^ has a faithful finite beta_j-invariant
  trace,
- N^G= C1.bbC1.
+ N^G= C1.C1.
 ]
 
 
@@ -1312,7 +1312,7 @@ the complementary regular representation obtained by omitting slot j;
 Lemma~lem:setup-normal-tensors makes it finite, with normalized
 beta_j-invariant trace since the remaining slot acts by unitaries in
 that commutant. Including all slots and using the last assertion of that
-lemma gives N^G= C1.bbC1.
+lemma gives N^G= C1.C1.
 
 
 ## The source groups and full versus partial mixing
@@ -1375,13 +1375,13 @@ groups and their root-generated lifts from
 Convention~conv:setup-panels [cite]; an independent
 proof in this convention is cite[Theorem~1.1 and Definition~4.23]{Ciobotaru2015}.
 We justify the type-I input for these actual groups. For an isotropic
-simple factor mathbf HbfH over k, choose a nonzero parabolic unipotent
-radical mathbf U.bfU. The k-span of the mathbf H(k)-conjugatesbfH(k)-conjugates of
-Lie(mathbf U)bfU) is a nonzero invariant Lie ideal, hence all
-of Lie(mathbf H).bfH). Choose X_1,...,X_d from these
+simple factor H over k, choose a nonzero parabolic unipotent
+radical U.U. The k-span of the H(k)-conjugatesbfH(k)-conjugates of
+Lie(U)U) is a nonzero invariant Lie ideal, hence all
+of Lie(H).H). Choose X_1,...,X_d from these
 conjugate subspaces; the map
 (t_1,...,t_d)|->prod_jexp(t_jX_j) has invertible differential
-at zero and lands in mathbf H(k)^+.bfH(k)^+. The local-field inverse function
+at zero and lands in H(k)^+.H(k)^+. The local-field inverse function
 theorem makes this subgroup open; at finite places see
 cite[Proposition~1.11]{Gloeckner2008}. The same holds for the root-generated
 subgroup of an algebraic simply connected cover. At an archimedean place
@@ -1395,9 +1395,9 @@ cite[Proposition~6.E.21(1)]{BekkaHarpe2019}.
 Restrict a separable unitary representation pi to one simple factor S.
 Its type-I decomposition with multiplicity is
 [
- H={widehatwidehat S}S^{ ⊕ }(H_xi ⊗  M_xi),deta(xi),
+ H={Shat}S^{ ⊕ }(H_xi ⊗  M_xi),deta(xi),
  
- pi(s)=int_{widehatwidehat S}S^{ ⊕ }(xi(s) ⊗ 1),deta(xi).
+ pi(s)=int_{Shat}S^{ ⊕ }(xi(s) ⊗ 1),deta(xi).
 ]
 In this central decomposition every unitary commuting with S acts
 fibrewise as 1 ⊗  V_xi
@@ -1433,10 +1433,10 @@ on a Cartan path exact root coordinates will determine which flag
 coordinates survive.
 
 An archimedean panel inside a higher-rank component cannot have noncompact
-Lie algebra mathfrak f_frakf_{4(-20)}: its complexification would have to
+Lie algebra f_{4(-20)}: its complexification would have to
 occur as a component of a proper absolute Dynkin subdiagram, and the
 Dynkin list has no such occurrence. At a finite place the effective
-rank-one Lie algebra is simple also over Q_p:bbQ_p: after restriction
+rank-one Lie algebra is simple also over Q_p:Q_p: after restriction
 of scalars its absolutely simple factors form one Galois orbit, so a
 Q_p-idealbbQ_p-ideal is either zero or the whole algebra. An adjoint
 automorphism centralizing an open subgroup centralizes all sufficiently
@@ -1453,7 +1453,7 @@ Write mu(g) for the Cartan vector only when its argument is a
 local-group element; this is distinct from the scalar two-cocycle
 mu(g,h). For a positive root alpha put
 [
- m_alpha=dim_{k}mathfrak g_frakg_alpha at a finite place,
+ m_alpha=dim_{k}g_alpha at a finite place,
  2rho=sum_{alpha>0}m_alphaalpha,
 ]
 and use the analogous real restricted-root multiplicities at archimedean
@@ -1651,7 +1651,7 @@ translation produces exactly one divergent target gap while successively
 completing such a vertex-origin path adds exactly one target gap per stage.
 
 We retain the regular realizations of Section~sec:setup: H=eL^2(B)^k,
-H=L^2(Y;H),calH=L^2(Y;H), d=(Tr ⊗ tau_B)(e). For an integrable corner
+H=L^2(Y;H),H=L^2(Y;H), d=(Tr ⊗ tau_B)(e). For an integrable corner
 field write
 [
  T(C)= ⊗ tau_B)(C(y)),dy,
@@ -1866,7 +1866,7 @@ is beta_j-equivariant, and is invariant under the source face parabolic.
 Its gap-stratum masses belong to N and are scalar. Conjugating
 the source split frame transports the measurement by sigma.
  If the path escapes in every simple component, the link is an isometry
-Hbar ⊗ H K.calK. Its output forms are
+Hbar ⊗ H K.K. Its output forms are
 decomposable at the two bases, covariant under the two independent target
 shifts, invariant under the independently contracted source radicals, and
 invariant under simultaneous source centralizers. An image-and-stratum
@@ -1941,7 +1941,7 @@ escaping sequence. By Howe--Moore applied to the representation on
 L^2(N), a radical-invariant vector orthogonal to the
 factor-invariant subspace would have a matrix coefficient both constant and
 tending to zero. It is therefore factor invariant, and the mass is invariant
-under all simple factors, hence scalar because N^G= C1.bbC1.
+under all simple factors, hence scalar because N^G= C1.C1.
 This argument covers a pure vertex and a mixed path alike.
 
 (ii) For bounded b,f, cyclicity and the definition of the induced
@@ -2109,7 +2109,7 @@ unchanged. Base decomposability and commutation with companions follow
 before limits. An invariant one-column test gives an element of
 N; face-parabolic invariance, followed by the active-factor
 radical argument and the inactive-factor centralizer argument of
-Proposition~prop:links-link, puts it in N^G= C1.bbC1.
+Proposition~prop:links-link, puts it in N^G= C1.C1.
 
 For mixed links, independent target invariance puts the respective slices
 in Nbar and N; independent
@@ -2251,7 +2251,7 @@ source action sigma|_S.
 
 Proof. 
 Suppose Phi:C(Z)toA_j is such a map. Its mass a=Phi(1)
-belongs to N^S:calN^S: target equivariance makes it commute with the
+belongs to N^S:N^S: target equivariance makes it commute with the
 measured target action, and membership in A_j the rest. Choose
 ε>0 with p=1_{[ε,∞)}(a)ne0. On pH,
 Psi(u)=pa^{-1/2}Phi(u)a^{-1/2}p is unital and completely positive (a
@@ -2270,8 +2270,8 @@ Use the minimal Stinespring construction with commutant lifting
 vectors u ⊗ xi with form
 <sum_iu_i ⊗ xi_i,sum_lv_l ⊗ eta_l>
 =sum_{i,l}<xi_i,Psi(u_ibarv_l)eta_l>, and
-widehatwidehat D(uD(u ⊗ xi)=u ⊗  Dxi satisfies
-|widehatwidehat DD|le|D| in the concrete input norm. The covariant lift
+Dhat(uD(u ⊗ xi)=u ⊗  Dxi satisfies
+|DDhat|le|D| in the concrete input norm. The covariant lift
 T_g(u ⊗ xi)=(g· u) ⊗  W_gxi has the same multiplier
 zeta as W.
 
@@ -2352,7 +2352,7 @@ Lemma~lem:links-obstruction. Thus DnePi.
 
 The marginal inclusion. Let Psi(F)=V_Ω^*M_FV_Ω be a
 minimal full-data dilation with image-flag map x:Ω_Dto X_D and
-minimal image dilation V:H K_X.calK_X. The generator map
+minimal image dilation V:H K_X.K_X. The generator map
 M_uVf|-> M_{ucirc x}V_Ω f preserves inner products and extends
 to an isometric inclusion i:K_X K_ΩcalK_Ω with
 iV=V_Ω, intertwining the image spectral measure, base multiplication,
@@ -2457,10 +2457,10 @@ formula and change of base z=b_sy show the inner full-data law is
 precomposed by U_{b_s}: Psi_s(F)=U_{b_s}^*Psi(F)U_{b_s}, with
 unchanged image marginal. Set T^{(s)}=widetilde U_{b_s}^*TU_{b_s}. The
 marginal lift fixes (x,l), so its fibre expression is
-(U_{b_s}^X)^*D_x(l)^{1/2}U_{b_s}^X applied to Vf; inequality
+(U_{b_s}^X)^*D_x(l)^{1/2}U_{b_s}^X applied to Vf; ineuality
 (eq:links-positive-window) persists. With J_0f(x,l)=Vf(x),
 B|->< Bf,J_0f> is continuous on E by
-(eq:links-intertwiner-frame). Weak decay and the positive inequality
+(eq:links-intertwiner-frame). Weak decay and the positive ineuality
 give
 0le√(c_C)< f,Psi_s(C)f>le< T^{(s)}f,J_0f>to0;
 density of bounded columns and 0lePsi_s(C)le1 extend the vanishing to
@@ -2596,7 +2596,7 @@ Common apartments and addition. On compact transverse patches, the
 two ideal faces admit common apartment or flat sections with origins in a
 bounded set. After erasing the bounded thickness, po=z+A+O(1) and
 q^{-1}o=z-B+O(1) with A,B positive and z bounded, so the vector
-distance is A+B+O(1); the vector-distance inequality under bounded
+distance is A+B+O(1); the vector-distance ineuality under bounded
 endpoint perturbations gives the Cartan addition. This uses the whole
 ambient product and no discreteness of a proper projection.
 
@@ -2691,7 +2691,7 @@ conjugated basis functions so its full weight has the form
 The strong order of sum and supremum follows from Fourier expansion on a
 slightly larger real box with uniform derivative bounds; at finite places
 a finite clopen mask basis replaces it. Minkowski, Plancherel, and the
-Hilbert ideal inequality give
+Hilbert ideal ineuality give
 |(sum_nc_n(k)[A_0(D_{hu_n}A_0)^*]_k)_k|_{ell^2(L^2)}
 le(sum_nsup_k|c_n(k)|)|A_0|_∞|D_hA_0|_2.
 The other side is symmetric. Cauchy--Schwarz in the shift variable then
@@ -2861,7 +2861,7 @@ all complementary factors, so all-factor mixing applies).
 Encode F_x by its minimal covariant Stinespring dilation and the vector
 obtained from the embedded columns of e, giving a positive functional
 phi_x on the separable matrix-amplified full twisted crossed product
-mathfrak CfrakC with norm d. The group and matrix operators act on these
+C with norm d. The group and matrix operators act on these
 columns by right multiplication; the regular-normal central summand of
 the group algebra bidual supplies canonical right multipliers, so the
 vector functional is supported on the corner of the corner, and the
@@ -2870,7 +2870,7 @@ equivariant subordinate maps (covariant Arveson domination
 cite[Theorem~1.4.2]{Arveson1969}).
 
 Fix the central support z of one functional of the second family.
-zmathfrak C^frakC^{**} has separable predual, and the components phi_x^z
+zC^C^{**} has separable predual, and the components phi_x^z
 form a measurable norm-separable family: test domination on a countable
 norm-dense positive family, let
 n(x)=sup{|theta|:0lethetalephi_x, theta is z-normal}
@@ -3050,7 +3050,7 @@ For the frame step: exhaust the compact windows to the event A of the
 chosen gap stratum and image condition I_i(O); monotone
 continuity gives E(A)J(barbar b,f)=0.b,f)=0. Then A is invariant under ignored
 left Fourier shifts, and the frame identity
-c_F< f,Psi_F(I_i(O))f>=calI_i(O))f>=frac1dsum_r
+c_F< f,Psi_F(I_i(O))f>=I_i(O))f>=frac1dsum_r
 |E(A)J(b_rbar,f)|^2=0 (with b_r the constant frame of e)
 combined with positivity and density gives (i). Arbitrary Borel output
 subprojections are imposed after buffered tests.
@@ -3118,7 +3118,7 @@ success languages in one direction; iterating p (finite order) closes
 the cyclic chain.
 
 
-[An inequality-detecting word]
+[An ineuality-detecting word]
 
 Every higher-rank component has a vertex type j and an incidence word
 that fails exactly when two initial type-j vertices are equal.
@@ -3130,11 +3130,11 @@ vertex of the opposite end type; success is equivalent to a hyperplane in
 the second point's position avoiding the first. For other crystallographic
 diagrams, choose a self-opposite leaf j with neighbor k and round
 trips from j to k and back; each permits the block
-W_{widehatwidehat j}j{1,s_j}W_{widehatwidehat j}j. With r the minimum number of
+W_{jhat}j{1,s_j}W_{jhat}j. With r the minimum number of
 s_j factor in a reduced longest word, r-1 round trips cannot end
 opposite from equality, and r can from the least nontrivial double
-coset W_{widehatwidehat j}js_jW_{widehatwidehat j}j. Every unequal position is above
-that least nontrivial double coset, so success at all unequal pairs
+coset W_{jhat}js_jW_{jhat}j. Every uneual position is above
+that least nontrivial double coset, so success at all uneual pairs
 follows from Lemma~lem:transfer-incidence-words.
 
 
@@ -3150,7 +3150,7 @@ measured copies.
 
 
 Proof. 
-Lemma~lem:transfer-inequality-word and its permuted inverse give
+Lemma~lem:transfer-ineuality-word and its permuted inverse give
 failure of the corresponding word at equality of source
 p^{-1}(j), so every ordered product of two candidates at that source
 label is supported on the target diagonal. The unital-POVM argument
@@ -3178,7 +3178,7 @@ For a general mixed face path of type I: the type-i source vertex z_i
 incident to F is unique for iin I. Proposition~prop:transfer-incidence(i)
 puts Psi_F(I_i(O)) inside P_{z_i}(O). A target face omitting
 p(i) admits two distinct incident vertices of that type, so disjoint
-O,O' make Psi_F(I_i(O) I_i(O'))calI_i(O')) supported in
+O,O' make Psi_F(I_i(O) I_i(O'))I_i(O')) supported in
 both P_{z_i}(O) and P_{z_i}(O'), hence zero. Countably many such
 pairs exhaust the omitted-type event, so every p(i) for iin I
 occurs. The upper bound of Proposition~prop:links-gap-count forces
@@ -3425,7 +3425,7 @@ Varying zeta, these intertwiners force the fibres d_t^+ to be graded
 by the normal representation types: each Schur subspace remains
 finite-dimensional per coset, giving a countable direct sum of finite
 matrix algebras. Ergodicity of T acting transitively on Ubackslash
-T forces D_*^T=C1:bbC1: any T-fixed normal functional on the
+T forces D_*^T=C1:C1: any T-fixed normal functional on the
 countable Cartesian product vanishes off the identity coset, and the
 resulting character is scalar.
 
@@ -3540,7 +3540,7 @@ G_n projects onto V, every block's stabilizer contains an analytic
 subgroup; a limiting source-only kernel would force one fixed open V_1
 into all late G_n's. Then [V_1,V_1] fixes D_n at every sufficiently
 late level, so [V_1,V_1]subsetLie V_1 satisfies
-[mathfrak v_1,frakv_1,mathfrak v_1]=frakv_1]=mathfrak v_1frakv_1 and hence V_1 is
+[v_1,v_1,v_1]=v_1]=v_1 and hence V_1 is
 nontrivial, in contradiction to the hypothesis.
 
 
@@ -3556,7 +3556,7 @@ Hence:
 
 [Dimensions and abelian corner]
 
-dim_{Q_p} s=fraks=dim_{Q_p} t.frakt. Every
+dim_{Q_p} s=s=dim_{Q_p} t.t. Every
 Hausdorff limit G_∞ has both projections isomorphisms, and the
 target-only subgroup G_∞cap(1times U) is {1}. Consequently
 D_0 is abelian and equals L^∞(U) with Haar probability.
@@ -3575,7 +3575,7 @@ in the subsequence; as D_n exhausts D_0, [V_1,V_1] fixes D_0,
 contradicting Lemma~lem:finite-limit-kernel. So Lie(G_∞)'s
 source-only part has zero Lie algebra. Both projections of
 Lie(G_∞) are then isomorphisms (surjections with equal dimension
-dimmathfrak s=fraks=dimmathfrak t).frakt). The target-only subgroup is a compact
+dims=s=dimt).t). The target-only subgroup is a compact
 zero-dimensional analytic subgroup of the torsion-free U, hence
 {1}.
 
@@ -3602,7 +3602,7 @@ left translation. In these coordinates, the commuting source action is
  
 
 where eta is a continuous homomorphism with isomorphic
-Q_p-differential.bbQ_p-differential.
+Q_p-differential.Q_p-differential.
 
 
 Proof. 
@@ -3643,7 +3643,7 @@ Its differential is injective: otherwise a small kernel V_1<V acts
 trivially on the center; since V fixes e, Equation~(eq:finite-center-corner)
 forces V_1 to act trivially on eD_*e, contradicting
 Lemma~lem:finite-limit-kernel. Since L is discrete and
-dimmathfrak s=fraks=dimmathfrak t,frakt, the closed Lie subgroup N_T(L) has
+dims=s=dimt,t, the closed Lie subgroup N_T(L) has
 dimension equal to T and is open. For lin L, conjugation from the
 open N_T(L) into the discrete L is continuous, so l centralizes an
 open subgroup; the open-centralizer assertion forces l=1. Thus
@@ -3653,13 +3653,13 @@ Remove matrix fibres: the type I decomposition and ergodicity of the
 center give D_*~= L^∞(T;B(V)) with constant separable
 dimension. In a measurable trivialization the action is
 (alpha_ad)(x)=c(a,x)d(a^{-1}x) with
-c(ab,x)=c(a,x)c(b,a^{-1}x) (an Aut(B(V))-calV))-
+c(ab,x)=c(a,x)c(b,a^{-1}x) (an Aut(B(V))-V))-
 valued cocycle). Changing variables and applying Fubini, freeze a generic
 x_0 and define b(x)=c(xx_0^{-1},x); then c(a,x)=b(x)b(a^{-1}x)^{-1}
 almost everywhere, and changing the fibre trivialization by b removes
 the cocycle. Pure translation fixes the constant field
 B(V); Equation~(eq:finite-fixed) forces dimmathcal
-V=1.calV=1.
+V=1.V=1.
 
 
 ## Ordinary homogeneous panel maps
@@ -3758,9 +3758,9 @@ on the panel type:
 
 with uniform error. On a dominant split translation the largest adjoint
 eigenvalue has logarithm a fixed positive multiple of the simple-root
-coordinate; converting from |·|_k to |·|_p^{[k:Q_p]}bbQ_p]
+coordinate; converting from |·|_k to |·|_p^{[k:Q_p]}Q_p]
 with the largest relative root multiple r_{max}in{1,2},
-d_R=r_{max}/[k:Q_p].bbQ_p]. Bounded centralizer and compact Cartan
+d_R=r_{max}/[k:Q_p].Q_p]. Bounded centralizer and compact Cartan
 factors change operator norms by bounded amounts. Symmetry under
 inversion gives d_R unchanged by opposition, so the d_R are drawn
 from a single type list.
@@ -3849,7 +3849,7 @@ measure.
 
 The horospherical group is N_b=Vtimes Z with
 [
- (v,z)(v',z')=(v+v',z+z'+tfrac12[v,v']),
+ (v,z)(v',z')=(v+v',z+z'+1/2[v,v']),
  delta_R(v,z)=(Rv,R^2z).
 ]
 Choose inner products with < J_zv,w>=< z,[v,w]>
@@ -3859,7 +3859,7 @@ N_btimes(0,∞), with a fixed normalization of Haar measure,
 
  ds_X^2=(dlambda^2)/(lambda^2)
        +(|dv|^2)/(4lambda^2)
-       +(|dz-tfrac12[v,dv]|^2)/(4lambda^4),
+       +(|dz-1/2[v,dv]|^2)/(4lambda^4),
   dp=frac{dn,dlambda}{lambda^{Q+1}}.
 
 These are the Iwasawa coordinates for the indicated symmetric spaces; the
@@ -4141,7 +4141,7 @@ finishes.
 
 ## The ray lift and its densities
 
-Let K_X=L^2(X,dp)calK_X=L^2(X,dp) and let T_g be translation on
+Let K_X=L^2(X,dp)K_X=L^2(X,dp) and let T_g be translation on
 K_X. Define
 [
  Delta:D→ B(K_X) ⊗⊗  D,
@@ -4181,12 +4181,12 @@ Delta on D and right density on D^{op}, both in L^2
 with norm at most C|Y|_∞|Y|_2; also
 |V_{d_1}Y(V_{d_1}Y)^*|_{2,Tr ⊗ tau} has this
 bound.
- If suppwidehatwidehat xxsubset{rle a} and a/d_1to0, then
+ If suppxxsubsethat{rle a} and a/d_1to0, then
 |V_{d_1}(xY)-Delta(x)V_{d_1}Y|_2to0 and
 |V_{d_1}(Yx)-(V_{d_1}Y)x|_2to0.
  For another long column Z and bounded x, if for every
-contributing triple hinsuppwidehatwidehat Y,Y, hkinsuppwidehatwidehat Z,Z,
-kinsuppwidehatwidehat xx one has
+contributing triple hinsuppYhat,Y, hkinsuppZhat,Z,
+kinsuppxxhat one has
 d_{∂ X}(h^+,(hk)^+)le e^{-cl'} with c>0,
 l'/d_1to∞, then
 
@@ -4228,7 +4228,7 @@ There are smooth annular partitions about g^+ at integer depths
 0le jle L (with exterior piece and innermost ball) on which
 
  
- |w_{d_1}(g,·)-a_{d_1}(j,L)|_{C^m_{rm normalized}}
+ |w_{d_1}(g,·)-a_{d_1}(j,L)|_{C^m_{normalized}}
  &le C_m/d_1,
  a_{d_1}(j,L)&=frac1{d_1}int_{s>j}psi(s/d_1)
                   psi((s-2j+L)/d_1),ds,
@@ -4257,7 +4257,7 @@ Delta_R=(1+R^2p(u))^2+R^4|W|^2,
 Q_R=(R^2(1+R^2p+R^2J_W)V/Delta_R,R^4W/Delta_R). Then
 
  beta=2j-L+B(R,t,u),
- B=tfrac12log(Delta_R/D_t),
+ B=1/2log(Delta_R/D_t),
  delta_Rn=m(R,t,u)=delta_{√(Delta_R)}(Q_R^{-1}P_t).
 
 All derivatives of B,m are bounded on the buffered annulus, and
@@ -4303,7 +4303,7 @@ Polarization extends to
  -< Z,xY>→0
 
 uniformly when norms of Z,Y,x are bounded with
-suppwidehatwidehat xxsubset{rle a}. Bounded approximants and
+suppxxsubsethat{rle a}. Bounded approximants and
 weakly convergent L^2 subsequences with slacked cutoffs handle
 infinite Fourier support.
 
@@ -4313,7 +4313,7 @@ K_{d_1}(v,v)=1; with d(o,p)le2d_1+C and fixed-derivative
 uniform bounds, Taylor at the diagonal on adjacent boxes of radius
 e^{-d_2} gives
 
- |K_{d_1}-1|_{C^m_{rm normalized}}le C_m e^{C_m d_1-d_2},
+ |K_{d_1}-1|_{C^m_{normalized}}le C_m e^{C_m d_1-d_2},
 
 positive-derivative terms contributing e^{-d_2}, central ones
 e^{-2d_2}. Choose d_1ll d_2ll l'. Separate K_{d_1}-1 at
@@ -4479,7 +4479,7 @@ A(sy)^* pairs against X^{Θ_n}_s(y)B(y)C(y,s); convert to
  tau(zeta_n(sy)^*Delta_T(X^{Θ_n}_s(y))xi_n(y)C(y,s))
 
 with xi_n=V_{d_n}B, zeta_n=V_{d_n}(A^*). The
-exact trace inequality (via Cauchy--Schwarz on the two density fields,
+exact trace ineuality (via Cauchy--Schwarz on the two density fields,
 each with uniformly bounded trace-Hilbert norm by
 Proposition~prop:ray-lift) is bounded by C c_0^k because the
 combined amplified action has no invariant vector (admissibility).
@@ -4491,7 +4491,7 @@ For a short change z, the error is at most
 
 Limit nto∞ then kto∞.
 
-For input values 0 and +∞, use triangle inequalities and
+For input values 0 and +∞, use triangle ineualities and
 short approximations with smooth tail cutoffs (Proposition~prop:ray-radial);
 the lifted shortening argument extends because it uses positive
 lower length bounds and a uniform deficit but no upper bound.
@@ -4499,12 +4499,12 @@ lower length bounds and a uniform deficit but no upper bound.
 The output-measurement identification uses the exact spectral
 argument: with a(s,t)=s+t on the compact square, product tests
 supported on rectangles whose closures lie in a^{-1}(F)^c have
-dense span in ran P_{rm in}(a^{-1}(F)^c), so
-E(F)J P_{rm in}(a^{-1}(F)^c)=0 and
-E(F)J=E(F)J P_{rm in}(a^{-1}(F)). A closed-set exhaustion
-K_j↑ F^c gives E(K_j)JP_{rm in}(a^{-1}(F))=0, so
-E(F^c)JP_{rm in}(a^{-1}(F))=0; together,
-E(F)J=J P_{rm in}(a^{-1}(F)). Monotone classes and spectral
+dense span in ran P_{in}(a^{-1}(F)^c), so
+E(F)J P_{in}(a^{-1}(F)^c)=0 and
+E(F)J=E(F)J P_{in}(a^{-1}(F)). A closed-set exhaustion
+K_j↑ F^c gives E(K_j)JP_{in}(a^{-1}(F))=0, so
+E(F^c)JP_{in}(a^{-1}(F))=0; together,
+E(F)J=J P_{in}(a^{-1}(F)). Monotone classes and spectral
 integration give
 E(phi)J=JM_{phi(s+t)} for phiin C([0,∞]).
 Convergence on the whole sequence follows by compactness of
@@ -4624,7 +4624,7 @@ Complementary tightness is the pure-ray conclusion of
 Section~sec:transfer, transported through the corner equivalence.
 Before the equivalence, complementary regularity supplies a nonzero
 constant frame column f for pi_0(Gamma) with W_Λ; if
-e_j are the finitely many identity columns, the frame inequality
+e_j are the finitely many identity columns, the frame ineuality
 applied to W_he_j gives, for every hinGamma,
 
  sum_{gammainGamma}
@@ -4686,7 +4686,7 @@ also has scalar relative commutant.
 Proof. 
 Smooth radial truncations are uniformly bounded (Proposition~prop:ray-radial),
 so tight elements admit Hilbert approximation by bounded sequences
-supported in r_Tle CL_n. Triangle inequality (eq:ray-triangle)
+supported in r_Tle CL_n. Triangle ineuality (eq:ray-triangle)
 gives closure under multiplication; length-invariance under inversion
 gives closure under adjoint; tightness is closed in Hilbert norm on
 each bounded ball.
@@ -4732,7 +4732,7 @@ construction < D_U,e_calU,e_{Q_0}> gives
 
 which is O(ε_K^2) by tightness plus smooth radial
 cutoffs (Proposition~prop:ray-radial). The Hilbert-space
-Kazhdan inequality supplies a positive contraction z close to
+Kazhdan ineuality supplies a positive contraction z close to
 e_{Q_0}; spectral rounding p_0=1_{[1/2,1]}(z) gives
 |p_0-e_{Q_0}|_2=O(ε_K)
 
@@ -4908,7 +4908,7 @@ D'_nto∞: feasibility is nonempty (identity embedding
 into L_eta(Gamma) suffices), so the infimum
 
  inf{tge1:|D_{1_{{r_i>L}}}Θ(gamma)|_2leε_K
- (gammainF), P_{Θ,D'_nL}(t)([1,∞])getfrac12}
+ (gammainF), P_{Θ,D'_nL}(t)([1,∞])ge1/2}
  
 
 is attained up to factor two at some witness. Lemma~lem:scale-tightening
@@ -5036,7 +5036,7 @@ give D_X(f)xiin L^2 on charts; a partition of unity globalizes.
 
 
 
-Set K=L^2(B_S,dx;calK=L^2(B_S,dx;H) and
+Set K=L^2(B_S,dx;K=L^2(B_S,dx;H) and
 C=span{x|-> a(x)P_x(g)xi:
 ain L^∞(B_S), gin C^∞(B_T), xiinE}. For real
 f and real horizontal X, the rule
@@ -5045,7 +5045,7 @@ f and real horizontal X, the rule
  =sum_ja_jP_bullet(g_j)D_X(f)xi_j
 
 defines a densely defined symmetric operator on C. If real
-smooth g_j,f_j satisfy sum_jg_j,d_{rm hor}f_j=0, then
+smooth g_j,f_j satisfy sum_jg_j,d_{hor}f_j=0, then
 
  sum_jP_bullet(g_j)T_X(f_j)=0quadhbox{on }C.
 
@@ -5071,13 +5071,13 @@ proves symmetry and, when a formal expression represents zero, shows
 its image is orthogonal to a dense subspace, hence zero. This gives
 well-definedness, symmetry, and closability. Cancellation: the kernel
 F(w,v)=sum_jg_j(w)(f_j(w)-f_j(v)) has order ge2 when
-sum_jg_jd_{rm hor}f_j=0, and the same limit shows
+sum_jg_jd_{hor}f_j=0, and the same limit shows
 sum_jP_x(g_j)D_X(f_j)(x)xi=0; commuting further multipliers
 proves (eq:scale-horizontal-cancellation).
 
 
 For a real smooth horizontal target covector field
-theta=sum_jg_jd_{rm hor}f_j, set T_X(theta)=sum_j
+theta=sum_jg_jd_{hor}f_j, set T_X(theta)=sum_j
 P_bullet(g_j)T_X(f_j); equation (eq:scale-horizontal-cancellation)
 makes this independent of expression. (Global finitely many target
 coordinate differentials span, and their Gram operator is smoothly
@@ -5098,7 +5098,7 @@ Proof.
 Choose finite smooth Parseval horizontal source fields (X_alpha)
 and target covectors (theta_r) (invert positive Gram operators on
 V_R). Define the finite column
-D:C K^calK^{n_Sn_T},
+D:C K^K^{n_Sn_T},
 Du=(T_{X_alpha}(theta_r)u)_{alpha,r}, closable by
 C density; set l=|overlineoverline D|.D|. Parameter and angular
 unitary multipliers commute with D on C, extending to
@@ -5174,7 +5174,7 @@ amplified domain. Finite-coordinate projections commute with the PVMs,
 preserve the domain, and commute with weak derivatives; on the
 algebraic tensor core C{e_n} the
 column is exactly D ⊗ 1, graph-densely. So
-l^{rm amp}=l ⊗ 1 and b^{rm amp}=b ⊗ 1; unitary
+l^{amp}=l ⊗ 1 and b^{amp}=b ⊗ 1; unitary
 identification with a normal copy preserves the identities, without
 any mixing on an enlarged commutant.
 
@@ -5326,7 +5326,7 @@ and the constants apply equally to the opposite orientation.
 
 Choose a target maximizing module V with largest split weight d_T,
 n_0=dim V, and (if a compact diagram symmetry interchanges
-inequivalent modules) replace V by the direct sum of its diagram
+ineuivalent modules) replace V by the direct sum of its diagram
 conjugates; every summand has the same optimal ratio, so still
 
  h(V)=n_0/c_T.