diff --git a/hard_sphere_boltzmann_fluctuations_v2.txt b/hard_sphere_boltzmann_fluctuations_v2.txt
index 466532f..760b5cb 100644
--- a/hard_sphere_boltzmann_fluctuations_v2.txt
+++ b/hard_sphere_boltzmann_fluctuations_v2.txt
@@ -149,11 +149,11 @@ Fix the sphere diameter ε>0 and set the mean density
 mu=ε^{-2}. The grand-canonical initial law, including N=0, is
 
 P_ε(N, d Z_N)
-=Z_ε^calZ_ε^{-1}(mu^N)/(N!)
-1 _{D_N^ε}calD_N^ε(Z_N)prod_{i=1}^N f_0(z_i) d z_i
+=Z_ε^Z_ε^{-1}(mu^N)/(N!)
+1 _{D_N^ε}D_N^ε(Z_N)prod_{i=1}^N f_0(z_i) d z_i
 
 with the non-overlap set
-D_N^ε=calD_N^ε={(z_i):|x_i-x_j|>ε, ine j} and
+D_N^ε=D_N^ε={(z_i):|x_i-x_j|>ε, ine j} and
 1leZ_εcalZ_εle e^mu. Given the configuration, spheres move by
 free flight and deterministic elastic collisions, defined almost surely
 by Deng--Hani--Ma, Proposition 1.2. There is no random collision rule and no
@@ -188,7 +188,7 @@ Deltaphi =phi(x,v')+phi(x,v_*')-phi(x,v)-phi(x,v_*)
 d v_* d omega
 
 C_t(phi,psi)
-=tfrac12int_{R^3times R^3times R^3timesS^2}
+=1/2int_{R^3times R^3times R^3timesS^2}
 B(v-v_*,omega)f_t(x,v)f_t(x,v_*)Deltaphi Deltapsi
 d x d v d v_* d omega
 
@@ -270,7 +270,7 @@ per-vertex logarithms.
 
 6. Exact centering transfer (the centering transfer, below). The
 discrepancy estimate, the fixed moments of N/mu, and H"older's
-inequality show that expectations agree to o(mu^{-1/2}), which is
+ineuality show that expectations agree to o(mu^{-1/2}), which is
 what transfers the limit with exact centering.
 
 This stage is independent of the dynamics. It does two things: prove the
@@ -291,7 +291,7 @@ be the set of set-partitions of V. The factorial cumulant
 inversion of the factorial hierarchy,
 
 F_{h,0}^{ε}(Z_h)
-=sum_{piinP([h])}calP([h])prod_{Binpi}c_{|B|,0}^{ε}(Z_B)
+=sum_{piinP([h])}P([h])prod_{Binpi}c_{|B|,0}^{ε}(Z_B)
 
 The one-point function is g_0=c_{1,0}^{ε}=F_{1,0}^{ε}. Set
 
@@ -342,12 +342,12 @@ lim_{εto0}sqrtmu(Epi_0^ε(phi)-int f_0phi)
 
 Proof.
 (i) Factorial bound and moments. For fixed Z_h, let
-Z_ε[Z_h]calZ_ε[Z_h] be the partition function of the remaining
+Z_ε[Z_h]Z_ε[Z_h] be the partition function of the remaining
 particles, still mutually excluding and additionally avoiding the balls
 B(x_j,ε). Directly from the grand-canonical law,
 
-F_{h,0}^{ε}(Z_h)= 1 _{D_h^ε}calD_h^ε(Z_h)prod_{j=1}^hf_0(z_j)
-(Z_ε[Z_h])/(calZ_ε[Z_h])/(Z_ε)calZ_ε)
+F_{h,0}^{ε}(Z_h)= 1 _{D_h^ε}D_h^ε(Z_h)prod_{j=1}^hf_0(z_j)
+(Z_ε[Z_h])/(Z_ε[Z_h])/(Z_ε)Z_ε)
 
 The ratio is le1, giving the factorial upper bound. For h=1 it is
 the probability that B(x_1,ε) is empty of particle centers; by the
@@ -364,18 +364,18 @@ U_n(Z_n)={H conn. on [n]}prod_{{i,j}in H}(-chi_{ij}) (nge2).
 
 The tree bound is
 
-|U_n(Z_n)|calU_n(Z_n)|lesum_{T tree}prod_{{i,j}inT}{ij}
+|U_n(Z_n)|U_n(Z_n)|lesum_{T tree}prod_{{i,j}inT}{ij}
 
 This is the Penrose forest scheme. Root a connected labelled graph at
 vertex 1, assign graph-distance levels, and let the parent of each
 vertex be its least-labelled neighbour in the preceding level. For a
 fixed resulting tree T, the graphs giving it are exactly
-T⊆calT⊆ H⊆ R(T), where R(T)
+T⊆T⊆ H⊆ R(T), where R(T)
 adds all same-level edges and every edge to a preceding-level vertex with
 label ge that of the chosen parent. These intervals partition all
 connected graphs, and summing one interval gives
 
-prod_{einT}-chi_e)prod_{ein R(T)∖calT)∖T}-chi_e),
+prod_{einT}-chi_e)prod_{ein R(T)∖T)∖T}-chi_e),
 
 whose second factor lies in [0,1], proving the tree bound.
 
@@ -387,12 +387,12 @@ classical systems, in Statistical Mechanics (Benjamin, 1967).
 
 Integrating a tree from its leaves, keeping one free vertex, gives
 
-int f_0^{ ⊗  n}|U_n|calU_n|le n^{n-2}b_ε^{n-1} (nge2)
+int f_0^{ ⊗  n}|U_n|U_n|le n^{n-2}b_ε^{n-1} (nge2)
 
 Hence the connected-graph (Mayer) series is absolutely convergent when
 emu b_ε|a|_∞<1, and by exponentiation/analytic continuation
 
-logZ_ε[a]=calZ_ε[a]=sum_{nge1}(mu^n)/(n!)intU_n(Z_n)
+logZ_ε[a]=Z_ε[a]=sum_{nge1}(mu^n)/(n!)intU_n(Z_n)
 prod_{j=1}^nf_0(z_j)a(z_j) d Z_n
 
 From the partition function to the cumulants. Apply
@@ -423,7 +423,7 @@ vertex. One auxiliary vertex touches at most 27 root components: pick
 one adjacent root from each; their mutual distances exceed ε, and
 their disjoint balls of radius ε/2 lie in a ball of radius
 3ε/2. Hence qle27n for a contributing tree with n auxiliary
-vertices. Using the counting inequality and d_ε=O(ε),
+vertices. Using the counting ineuality and d_ε=O(ε),
 
 frac{|c_{h,0}^{ε}(Z_h)|}{prod_jf_0(z_j)}
 le(Ch)^{Ch}ε^{q/27}le(Ch)^{Ch}ε^{(q-1)/81} (qge2),
@@ -578,7 +578,7 @@ initial form has the corresponding bound.
 
 Let I_0,I be independent isonormal Gaussians on L^2(f_0 d z) and
 L^2(nu) with
- d nu=tfrac12B(v-v_*,omega)f_r(x,v)f_r(x,v_*) d r d x d v d v_* d omega.
+ d nu=1/2B(v-v_*,omega)f_r(x,v)f_r(x,v_*) d r d x d v d v_* d omega.
 Define
 
 zeta_t(h)=I_0(U(0,t)h)+I( 1 _{rle t}Delta_{U(r,t)h}),
@@ -720,15 +720,15 @@ Integral convention. Give each bond e a vector z_e=(x_e,v_e)
 with position X_e(t)=x_e+tv_e; give atom a a time t_a. Its incoming
 edges are 1,2 and serial outgoing edges 1',2'. On the contact surface
 put u=v_1-v_2, omega=(X_1(t_a)-X_2(t_a))/ε, and
-mathfrak b=[-ufrakb=[-u·omega]_+. Define
+b=[-ufrakb=[-u·omega]_+. Define
 
-Delta_a^{mathrm C}rmC
-=delta(|X_1-X_2|-ε)mathfrak bfrakb
+Delta_a^{C}rmC
+=delta(|X_1-X_2|-ε)b
 prod_{j=1}^2delta(X_{j'}(t_a)-X_j(t_a))
 delta(v_{1'}-v_1+(u·omega)omega)delta(v_{2'}-v_2-(u·omega)omega)
 
-Delta_a^{mathrm O}rmO
-=delta(|X_1-X_2|-ε)mathfrak bfrakb
+Delta_a^{O}rmO
+=delta(|X_1-X_2|-ε)b
 prod_{j=1}^2delta(x_{j'}-x_j)delta(v_{j'}-v_j)
 
 Reverse omega to match the scattering convention; surface measure is
@@ -756,7 +756,7 @@ requirement; it may have up to simΛ_i^2 lines.
 T_i^{err} inserts one extra O-atom into a member of
 T_i. The one-root signed tree sum contains every one-root
 particle-line tree of size <Λ_i/3, with C/O choices and sign
-(-1)^{#mathrm O}rmO.
+(-1)^{#O}rmO.
 
 Theorem (Deng--Hani--Ma analytic package). Assume the kinetic assumption, the cutoff hierarchy, the exponent hierarchy, and a common Gaussian Bol bound on all birth
 inputs. For L sufficiently large (fixed) and ε sufficiently small:
@@ -775,7 +775,7 @@ only satisfy the common Gaussian Bol envelope.
 |g-f_{(i-1)tau}|_{Bol,beta_{i-1}}leε^{theta_{i-1}}, then the
 signed tree sum T_i^ε[g]sfT_i^ε[g] satisfies
 
-|T_i^ε[g]-f_sfT_i^ε[g]-f_{itau}|_{Bol,beta_i}letfrac12ε^{theta_i}
+|T_i^ε[g]-f_sfT_i^ε[g]-f_{itau}|_{Bol,beta_i}le1/2ε^{theta_i}
 
 In the fixed-top-endpoint norm Bol,widetildebeta_i, the summed
 absolute contributions of cyclic T_i, of
@@ -847,7 +847,7 @@ root lines), provided the root cutoffs have the corresponding slack.
 Bounded records carried by these lines are kept.
 
 Proof (Source and construction).
-(i): use the error form of the cutting inequality Deng--Hani--Ma, Eq. (9.51)
+(i): use the error form of the cutting ineuality Deng--Hani--Ma, Eq. (9.51)
 retaining its deficit propto h. The finale Deng--Hani--Ma, Section 13.5, p. 169 splits
 into a bounded-rho case, where the UP algorithm makes le h free-
 component ``bad pieces'' paid by the root deficit, and a large-rho case
@@ -871,7 +871,7 @@ Insert once
 per continuation: the dummy activity mu cancels the extra root factor
 mu^{-1}, its straight line satisfies continuation in every slab, and
 the new links are pendant so the initial forest is preserved. This is a
-positive kernel inequality, valid even where f_0 vanishes; the envelope
+positive kernel ineuality, valid even where f_0 vanishes; the envelope
 extension of the analytic package admits q_0, and records are
 untouched.
 
@@ -914,12 +914,12 @@ where all finite subsystems are defined.
 
 Lemma (Autonomous partition). For finite V,
 
-sum_{C P(V)}calP(V)prod_{CinC}
+sum_{C P(V)}P(V)prod_{CinC}
 prod_{{C,D}⊆C},D)=1
 
 For S⊆ V and a root test H,
 
-H(Psi_V|_S)=sum_{C P(V)}calP(V)
+H(Psi_V|_S)=sum_{C P(V)}P(V)
 prod_{CinC}{{C,D}⊆C},D)
 H(Psi_{C}|_S)
 
@@ -945,10 +945,10 @@ There is a finite signed remainder R with
 
 prod_{e⊆ W}(1-a_e)
 =sum_{R⊆ I⊆ W}
-sum_{substack{B P(I)calP(I) Bcap Rne∅, |B|<M (BinB)}}
+sum_{substack{B P(I)P(I) Bcap Rne∅, |B|<M (BinB)}}
 prod_{BinB}{e⊆ W∖ I}(1-a_e)+R
 
-|R|calR| is bounded by a sum of rooted forests, each with exactly one
+|R|R| is bounded by a sum of rooted forests, each with exactly one
 component of size in [M,2M-2] and all other components smaller than M,
 all meeting R, vertices off the forest keeping their mutual compatibility
 product, each selected forest with coefficient le1. Moreover
@@ -967,7 +967,7 @@ components of size <M, giving the single large component in [M,2M-2]
 and the asserted multiplicity. Completed branches produce all graphs with
 unique exploration, grouped by their reached components into products of
 u(B) with survivors keeping their compatibility product, giving the main
-term; stopped branches give the forest bound. For the tree inequality, fix
+term; stopped branches give the forest bound. For the tree ineuality, fix
 a total edge order on B, assign each connected graph its minimum spanning
 tree T, let Q(T) be the edges later than the whole T-path; then
 connected graphs on B are exactly T⊆ G⊆ Tcup Q(T), so
@@ -990,8 +990,8 @@ F_s=mu^{-s}sum_{nge s}(1)/((n-s)!)(pr_{[s]})_#p_n
 For S=[s], V=[s+j], define the signed root-output kernel
 
 K_{S,V}(X, d X'_S)
-=sum_{C P(V)}calP(V)prod_{CinC}
-sum_{substack{B P(calP(C)
+=sum_{C P(V)}P(V)prod_{CinC}
+sum_{substack{B P(P(C)
 (∪_{Cin B}C)cap Sne∅, |B|<M}}
 prod_{BinB};X) delta_{Psi_{C}|_S}( d X'_S)
 
@@ -1004,8 +1004,8 @@ int F_{s+j}( d X_V) K_{S,V}(X, d X'_S)
 (An empty restriction is zero; at most s retained blocks, so the bound on
 j follows from the caps.) The stopped kernel replaces the inner
 block sum by
-sum_{Finmathfrak F^frakF^{stop}_S(C)}{ein E(F)}a_e(X),
-where mathfrak F^frakF^{stop}_S(C) are forests spanning
+sum_{FinF^F^{stop}_S(C)}{ein E(F)}a_e(X),
+where F^F^{stop}_S(C) are forests spanning
 C with every component meeting S, exactly one of size
 [M,2M-2], the rest smaller, no unrooted isolated cluster; keep
 prod_Ckappa_C and the root-output map. Define E_sF by
@@ -1017,7 +1017,7 @@ Lemma (One-slab identity and variation remainder). For a positive input law with
 hierarchy after the truncated slab. If factorial measures up to
 the fresh-label bound are finite, then
 
-F'_s=L_sF+R_s,calL_sF+R_s, |R_s|leE_sF
+F'_s=L_sF+R_s,L_sF+R_s, |R_s|leE_sF
 
 The record version holds, and E_sF is a positive domination of
 R_s, not a claim that R_s has this local kernel.
@@ -1032,7 +1032,7 @@ frac1{(n-s)!}binom{n-s}{j}=frac1{j!(n-s-j)!} turns the remaining sum
 into mu^{s+j}F_{s+j}; the output normalization mu^{-s} leaves
 mu^j/j!. Stopped terms keep only the unreached compatibility product and
 drop the rest, so positivity of the input and the multiplicity bound give
-|R_s(H)|le(E_sF)(|H|);calE_sF)(|H|); there are le s rooted components and
+|R_s(H)|le(E_sF)(|H|);E_sF)(|H|); there are le s rooted components and
 only one of size up to 2M-2 clusters, giving
 the fresh-label bound. Records are untouched by the spectator
 summation and travel with the root-output map.
@@ -1049,7 +1049,7 @@ activity lemu^q, and at most A_iΛ_i^2 values of q; take logs.
 Define the unmarked one-root measures recursively by
 
 g_i=U_i(g_{i-1}),
-U_i(g):=calU_i(g):=L_{i,1}((g^{ ⊗  q})_{qge0})
+U_i(g):=U_i(g):=L_{i,1}((g^{ ⊗  q})_{qge0})
 
 a finite signed measure using the full retained one-root block. It
 approximates the Boltzmann density:
@@ -1080,7 +1080,7 @@ between two flight segments.
 
 Families (i)--(ii) are subsets of the DHM one-root class; (iii) is its
 additional-overlap class. Coefficients carry only a fixed cutoff power, and
-the leading tree coefficient is exactly (-1)^{#mathrm O}rmO with the
+the leading tree coefficient is exactly (-1)^{#O}rmO with the
 labelled factorial mu^{n-1}/(n-1)!.
 
 Proof.
@@ -1093,7 +1093,7 @@ collisions, so each pair of straight segments has le1 incoming contact;
 partition the encounter event by the first admissible segment pair, giving
 disjoint encounter indicators with weights in [0,1]. Ordering these along
 lines gives particle-line molecules. C-clusters plus the O forest determine
-the connected block with sign (-1)^{#mathrm O}rmO and global excess equal
+the connected block with sign (-1)^{#O}rmO and global excess equal
 to the sum of C-cluster excesses, so cyclic terms and ngeΛ_i/3
 terms are exactly (i)--(ii), inside the DHM caps Λ_i<∞,
 Gamma. A remaining tree molecule with no other encounters has autonomous
@@ -1123,13 +1123,13 @@ coordinates on R. The operators L_i append the slab
 observations on the same labels; compatibility is record-independent. The
 retained hierarchy is the finite composition
 
-widehatwidehat F^R_i=F^R_i=L_i L_calL_{i-1}...L_1F^R_0
+Fhat^R_i=F^R_i=L_i L_calL_{i-1}...L_1F^R_0
 
 well defined order by order (finitely many previous orders per output).
 
 Lemma (Resummation with records). There is an exact finite expansion
 
-widehatwidehat F^R_F^R_{i,S}=sum_{R⊆ H⊆ S}E^R_{i,H} ⊗  g_i^{ ⊗ (S∖ H)}
+Fhat^R_F^R_{i,S}=sum_{R⊆ H⊆ S}E^R_{i,H} ⊗  g_i^{ ⊗ (S∖ H)}
 
 The E^R_{i,H} are represented by layered diagrams: every retained block
 in slab i contains a root of the interface above; a block is retained if
@@ -1164,7 +1164,7 @@ the exact blocks; artificial links for marked lines are not connections.
 Lemma (Labels and connected components). The retained expansion factors over global components. If D^R_{i,B} is
 the sum of diagrams connected on roots B, then
 
-widehatwidehat F^R_F^R_{i,S}=sum_{piinP(S)}⊗_{Binpi}D^{Rcap B}_{i,B}
+Fhat^R_F^R_{i,S}=sum_{piinP(S)}⊗_{Binpi}D^{Rcap B}_{i,B}
 
 Hence factorial-cumulant inversion keeps exactly the connected diagrams,
 even after resumming into g_i. At fixed root order h=|S| and bounded
@@ -1200,14 +1200,14 @@ exponent controlled by crossings and initial links.
 Lemma (Stopped variation errors). For L large in terms of the fixed data and Gamma, any fixed record
 bound, and |S|le A_i/2, there are c_i>0 with
 
-|F^R_{i,S}-widehatwidehat F^R_F^R_{i,S}|_{TV}le e^{-c_iΛ_i}
+|F^R_{i,S}-Fhat^R_F^R_{i,S}|_{TV}le e^{-c_iΛ_i}
 
 uniformly for a partial final slab (no velocity weight claimed).
 
 Proof.
-Write Delta_i=F_i-widehatwidehat F_i.F_i. Then
+Write Delta_i=F_i-Fhat_i.F_i. Then
 Delta_i=R_i+L_i{i-1}, |R_i|leE_iF_{i-1},
-Delta_0=0. Substitute F_{i-1}=widehatwidehat F_F_{i-1}+Delta_{i-1} in the
+Delta_0=0. Substitute F_{i-1}=Fhat_F_{i-1}+Delta_{i-1} in the
 positive stopped bound and expand the retained part as in
 the resummation with records; refining autonomous histories into C graphs
 and picking one incoming contact per incompatibility gives exactly the
@@ -1663,7 +1663,7 @@ once ε is small; by the no-extra-contact lemma almost every merger parameter
 set has only prescribed incoming contacts for all small ε. Thus every
 acceptance restriction holds, counting an encounter or its existence is the
 same, and the connected compatibility coefficient equals its tree sign
-(-1)^{#mathrm O}rmO. Replacing g_i by f_{itau} in a fixed-size term is
+(-1)^{#O}rmO. Replacing g_i by f_{itau} in a fixed-size term is
 justified by a telescoping product and the Gaussian suppression, with
 the uniform summability bound (one factor replaced by its Bol-difference)
 giving convergence to zero. Dominated convergence in space (checked first
@@ -1882,7 +1882,7 @@ integral, with  d x d omega.
 
 Proposition (Evolution of the marked tree laws). Between updates,
 
-∂_tF_t(Phi) =F_t(DPhi)+tfrac12int BF_tF_{t,*}DeltaPhi
+∂_tF_t(Phi) =F_t(DPhi)+1/2int BF_tF_{t,*}DeltaPhi
 
 ∂_tJ_t(Phi ⊗ Psi) =J_t((D+H_t)Phi ⊗ Psi+Phi ⊗ (D+H_t)Psi)
 
@@ -1935,7 +1935,7 @@ The pre/post substitution with omega|->-omega preserves
 B(v-v_*,omega) d v d v_* d omega, as does particle interchange with
 the normal reversed. Hence the normalization in the Boltzmann equation gives
 
-int Q(f,f)phi=tfrac12int Bff_*Deltaphi,
+int Q(f,f)phi=1/2int Bff_*Deltaphi,
 int[Q(g,f)+Q(f,g)]phi=int g K_tphi
 
 with background f=f_t. So the collision part of the linearized weak drift
@@ -1948,12 +1948,12 @@ G_t(Phi,Psi)=J_t(Phi ⊗ Psi)+F_t(PhiPsi)
 Then the tree-law evolution gives
 
 ∂_tG_t(Phi,Psi)=G_t((D+H_t)Phi,Psi)+G_t(Phi,(D+H_t)Psi)
-+tfrac12int BF_tF_{t,*}DeltaPhi DeltaPsi
++1/2int BF_tF_{t,*}DeltaPhi DeltaPsi
 
 Direct check of the source: with P=Phi+Phi_*, R=Psi+Psi_*, the
 collision change of the diagonal products from the F equation plus the
-J source, symmetrized, is tfrac12int BFF_*Delta(PR); the diagonal
-drifts contribute tfrac12int BFF_*(RDelta P+PDelta R); the difference
+J source, symmetrized, is 1/2int BFF_*Delta(PR); the diagonal
+drifts contribute 1/2int BFF_*(RDelta P+PDelta R); the difference
 is the last term of the G evolution since
 Delta(PR)=RDelta P+PDelta R+Delta P Delta R. This uses neither
 detailed balance nor equilibrium. The source is positive semidefinite
@@ -1976,7 +1976,7 @@ and conserved at a collision, so the marginal statement after
 the collision-position integral gives
 
 (D+H_t)(c y+h)=A_th,
-tfrac12int BF_tF_{t,*}Delta(c y+h)Delta(d y+k)=C_t(h,k)
+1/2int BF_tF_{t,*}Delta(c y+h)Delta(d y+k)=C_t(h,k)
 
 At pulse t_j, backward substitution sends c y+h to c y+h+cphi_j.
 
@@ -2023,7 +2023,7 @@ Put S^ε=sum_iy_i(T). Since |S^ε|le M_{obs}N and
 Z_εcalZ_εle e^mu, S^ε has exponential moments for each fixed
 ε. The exact factorial-to-ordinary identity at the final marked time is
 
-kappa_k(S^ε)=sum_{piinP([k])}calP([k])mu^{|pi|}
+kappa_k(S^ε)=sum_{piinP([k])}P([k])mu^{|pi|}
 int c_{|pi|}^ε( d xi_1... d xi_{|pi|})prod_{Binpi}y_B^{|B|}
 
 Directly: write
@@ -2209,7 +2209,7 @@ full graph (including chains through exceptional atoms), and let
 T_i be its time-order region. On the original support with total
 birth energy le P, an empty slab contributing one,
 
-|T_i|calT_i|supprod_{a ordinary in i}|v_a-v_{*a}|
+|T_i|T_i|supprod_{a ordinary in i}|v_a-v_{*a}|
 lefrac{tau^{q_i}(2q_iP)^{q_i/2}}{q_i!}
 
 Proof.
@@ -2248,7 +2248,7 @@ the surviving mass by prod_ell H_ell, and intervening deterministic
 substitutions have norm at most one on bounded functions. Nonnegative
 Fubini represents nu as the product in θ times these
 kernels, and majorizing the indicator of E by
- 1 _{D}θ) proves the extraction inequality.
+ 1 _{D}θ) proves the extraction ineuality.
 Exhaustion removes the boxes. Equivalently, restrict ordinary relative
 speeds away from zero and bound a rectangle by its product volume; cover a
 compact containing set by products of rectangles and take the infimum;
@@ -2352,7 +2352,7 @@ neutral type-two piece and a singleton with the same bound.
 The bounds are uniform in previously exposed lines and times. This is
 (Deng--Hani--Ma, Lemma 15.12).
 
-Fix delta=tfrac12min{υ,1/20}>0; every retained local gain is
+Fix delta=1/2min{υ,1/20}>0; every retained local gain is
 at least ε^delta up to a fixed polylogarithmic factor.
 
 Lemma (Amplified pure-collision cut). For large dimensional Gamma_0, every connected realizable full scattering