diff --git a/log_sobolev_subgaussian_v2.txt b/log_sobolev_subgaussian_v2.txt
index 848c2d9..fd4d35f 100644
--- a/log_sobolev_subgaussian_v2.txt
+++ b/log_sobolev_subgaussian_v2.txt
@@ -1287,10 +1287,10 @@ belong to Dom(A_eta) in both branches. The choices are:
 
   strict branch:  eta = f^2 mu_s,   w = D log f^2,   I = 2 lambda H(eta | mu_s);
 
-  gap branch:  eta = mu_s,   w = D varphi,
+  gap branch:  eta = mu_s,   w = D phi,
 
-where varphi is a centered eigenfunction of A_{mu_s} with eigenvalue
-lambda, and I = lambda E_{mu_s} varphi^2.
+where phi is a centered eigenfunction of A_{mu_s} with eigenvalue
+lambda, and I = lambda E_{mu_s} phi^2.
 
 Proof. Strict branch. Lemma 10 supplies the structural identities and
 the curvature-mean inequality. Apply Lemma 11 to sigma = eta with
@@ -1313,30 +1313,30 @@ Gap branch. Compact form embedding supplies a minimizer of the
 Rayleigh quotient for A_{mu_s} among mean-zero unit-norm functions.
 Its positive eigenvalue is 1 / R_s^P = lambda. The nullspace is the
 constants, so the quotient has a positive gap. Interior elliptic
-regularity makes its eigenfunction varphi smooth. Commuting one
+regularity makes its eigenfunction phi smooth. Commuting one
 derivative gives
 
-  A_{mu_s} D varphi = (lambda Id - K_s) D varphi.
+  A_{mu_s} D phi = (lambda Id - K_s) D phi.
 
-The right side is in L^2(mu_s), so Lemma 3 puts D varphi in the
-operator domain. Pairing this display with varphi and applying
+The right side is in L^2(mu_s), so Lemma 3 puts D phi in the
+operator domain. Pairing this display with phi and applying
 Poincare at mu_s reproduces the curvature-mean inequality
 E_eta <w, K_s w> <= lambda |E_eta w|^2.
 
 The coordinate function Y also sits in the operator domain, because
 both Y and A_{mu_s} Y = (Y - b_s)/s lie in L^2(mu_s). Pairing
-A_{mu_s} varphi = lambda varphi against Y, using self-adjointness,
+A_{mu_s} phi = lambda phi against Y, using self-adjointness,
 gives
 
-  E_eta w = lambda E[varphi Y]
-          = (E[varphi Y] - E[varphi b_s]) / s,
+  E_eta w = lambda E[phi Y]
+          = (E[phi Y] - E[phi b_s]) / s,
 
-  (1 - lambda s) E[varphi Y] = E[varphi b_s].
+  (1 - lambda s) E[phi Y] = E[phi b_s].
 
 Conditional Jensen and Lemma 6 give E_{mu_s} b_s b_s^* <=
 E_mu X X^* <= C a^2 Id. For every unit vector v, Cauchy-Schwarz
-bounds |E[varphi v.b_s]| by C a ||varphi||_2, with no factor from
-dimension. Since 1 - lambda s >= 1/2 and I = lambda ||varphi||_2^2, the
+bounds |E[phi v.b_s]| by C a ||phi||_2, with no factor from
+dimension. Since 1 - lambda s >= 1/2 and I = lambda ||phi||_2^2, the
 display gives |E w| <= C_J a sqrt(I), and K_s^2 <= s^{-1} K_s
 finishes the uniform smallness of K_s w.
 
@@ -1386,10 +1386,10 @@ differentiability point of R_s the two-sided contact gives
 Q'(s) = R_s'. The energy identity and w = 2 Df / f give
 R_s' = R_s E<w, K_s w> / I.
 
-Gap branch. Take g = varphi and Q(t) = Var_{mu_t}(varphi_t) /
-int |D varphi_t|^2 dmu_t. Now Q(t) <= R_t^P <= R_t and
+Gap branch. Take g = phi and Q(t) = Var_{mu_t}(phi_t) /
+int |D phi_t|^2 dmu_t. Now Q(t) <= R_t^P <= R_t and
 Q(s) = R_s^P = R_s. The same contact argument gives the identical
-derivative formula with w = D varphi.
+derivative formula with w = D phi.
 
 Lemma 12 bounds the derivative by C_J sqrt(a) in either branch, with
 constants independent of which branch holds at each time. Lipschitz
@@ -1429,8 +1429,8 @@ subsequence.
       identity of Lemma 10 holds for compact square-root tests
       relative to eta, and A Y = (Y - b_s)/s - w.
 
-  (ii) R_s = R_s^P: eta = mu_s, A varphi = lambda varphi,
-       E_eta varphi = 0, w = D varphi, I = lambda E varphi^2.
+  (ii) R_s = R_s^P: eta = mu_s, A phi = lambda phi,
+       E_eta phi = 0, w = D phi, I = lambda E phi^2.
        Coefficients may be scaled positively so that I -> 0.
 
 All o-terms refer to the sequence, with all finite auxiliary parameters
@@ -1450,7 +1450,7 @@ The energy pairing yields
 and Poincare with constant lambda^{-1} is the asserted spectral gap.
 The coordinate assertion and the branch-specific identities are
 already proved. In the gap branch every estimate is homogeneous in
-varphi, so positive rescaling enforces I -> 0 while leaving each
+phi, so positive rescaling enforces I -> 0 while leaving each
 normalized claim intact. QED.
 
 
@@ -1468,8 +1468,8 @@ sequence index. Then
 
 Strict branch: eta = f^2 mu_s, w = D log f^2, 2 H(eta | mu_s) = I / lambda.
 
-Gap branch: eta = mu_s, w = D varphi, A_{mu_s} varphi = lambda varphi,
-E_{mu_s} varphi = 0, I = lambda E_{mu_s} varphi^2.
+Gap branch: eta = mu_s, w = D phi, A_{mu_s} phi = lambda phi,
+E_{mu_s} phi = 0, I = lambda E_{mu_s} phi^2.
 
 Every parameter T, M, r, epsilon is fixed before the sequence limit.
 
@@ -1871,7 +1871,7 @@ strict-branch assertion for every fixed T.
 
 Gap branch, without additional probes. Linearize Lemma 16. Let q be a
 bounded smooth centered test under mu_s with bounded gradient, and put
-d eta_eps = (1 + eps q) d mu_s with |eps| ||q||_infty < 1. This
+d eta_eps = (1 + eps q) d mu_s with |eps| ||q||_infinity < 1. This
 alternative has finite exponential-square moment. Write
 qbar_r = E_{mu_s}[q(Y_0) | U_r] and
 Dqbar_r = E_{mu_s}[D q(Y_0) | U_r]. The output density ratio is
@@ -1890,7 +1890,7 @@ together with its output version, gives
 The denominator is bounded away from zero uniformly in r, and
 conditional Jensen bounds the integrand by a constant times ||Dq||_2^2,
 justifying the limit. Approximation in the form norm extends (7.23) to
-q = varphi: centered compact smooth approximants are bounded, and
+q = phi: centered compact smooth approximants are bounded, and
 conditional expectation is L^2-contractive. To retain compact support
 while centering, subtract the approximant's mean times one fixed
 compact smooth function of mean one; that correction tends to zero in
@@ -1909,7 +1909,7 @@ calculation with the square function gives
   (d/dr - Delta/2)(mu_{s+r} |g_r|^2) = - mu_{s+r} |D g_r|^2.
 
 Integrate against expanding cutoffs; their Laplacians vanish uniformly.
-The bound |g_r| <= ||g||_infty removes the cutoff terms and gives the
+The bound |g_r| <= ||g||_infinity removes the cutoff terms and gives the
 integrated identity on positive time. The initial endpoint follows from
 ||g_r(U_r) - g(Y_0)||_2 -> 0: for bounded continuous g use g(U_r) as a
 predictor and dominated convergence under the coupling
@@ -1924,14 +1924,14 @@ Poincare at mu_{s+r} and preservation of the mean therefore give
 
   E_{mu_{s+r}} |g_r|^2 <= theta(r) E_{mu_s} |g|^2.
 
-Apply this first to varphi and then to w - E w. The second and (7.1)
-give m(r) <= theta(r) + o(1). The first and ||varphi||_2^2 = I/lambda,
+Apply this first to phi and then to w - E w. The second and (7.1)
+give m(r) <= theta(r) + o(1). The first and ||phi||_2^2 = I/lambda,
 inserted into (7.23), give
 
   int_0^z m(r) dr >= (1 - theta(z))/lambda.
 
 The same squeeze and monotonicity argument prove (7.20). Multiplying
-varphi by positive scalars multiplies every squared quantity by the
+phi by positive scalars multiplies every squared quantity by the
 square, so their choice arranges I -> 0. QED.
 
 
@@ -2068,7 +2068,7 @@ standard Gaussian independent of Y_0. Write
   h_r(y) = E[w(Y_0) | U_r = y],   h_0 = w,
   m(r) = ||h_r||_2^2 / I.
 
-Then m: [0, infty) -> [0, 1] is nonincreasing, and by Proposition 17
+Then m: [0, infinity) -> [0, 1] is nonincreasing, and by Proposition 17
 
   lim m(r) = e^{-r}   for every fixed r in (0, 1/4).   (8.1)
 
@@ -2158,9 +2158,9 @@ E[Z u], and summing across coefficients gives
 (b) Let E_S(mu) denote the spectral resolution of A applied to S. The
 identities
 
-  ||S||_2^2 = int_0^infty d||E_S(mu) S||_2^2,
+  ||S||_2^2 = int_0^infinity d||E_S(mu) S||_2^2,
 
-  ||DS||_2^2 = int_0^infty mu d||E_S(mu) S||_2^2,
+  ||DS||_2^2 = int_0^infinity mu d||E_S(mu) S||_2^2,
 
 together with (8.10), give
 
@@ -2172,7 +2172,7 @@ Spectral mass at mu = 0 carries weight |E S|^2 = o(I), and (8.12)
 bounds the mass of the remaining contribution at mu >= 1 + delta by
 o(I)/delta for every fixed delta > 0. Splitting the multiplier
 e^{1-mu} - 1 into the ranges {mu = 0}, [lambda, 1 + delta],
-(1 + delta, K], (K, infty) for large K: the first and fourth give
+(1 + delta, K], (K, infinity) for large K: the first and fourth give
 o(I), the second is O(delta) by |e^{1-mu} - 1| <= C |mu - 1| near
 mu = 1, and the third is at most sqrt(o(I)/delta). Sending delta -> 0
 after the sequence limit, with K free, proves ||S~ - S||_2 = o(sqrt I).
@@ -2183,12 +2183,12 @@ For the derivative norm,
 
 On [lambda, 1 + delta] the factor mu |e^{1-mu} - 1|^2 is at most
 C delta^2. On (1 + delta, K] it is at most K, and the mass there is
-bounded by o(I)/delta. On (K, infty) it is at most mu, and
+bounded by o(I)/delta. On (K, infinity) it is at most mu, and
 int_{mu > K} mu d||E_S S||_2^2 = o(I) by (8.12) with lambda -> 1.
 This proves ||D(S~ - S)||_2 = o(sqrt I).
 
 The spectral multiplier of (A - 1) S~ is (mu - 1) e^{1 - mu}, vanishing
-at mu = 1 and bounded on [0, infty). The same piecewise splitting
+at mu = 1 and bounded on [0, infinity). The same piecewise splitting
 gives
 
   int |(mu - 1) e^{1-mu}|^2 d||E_S S||_2^2
@@ -2359,7 +2359,7 @@ Splitting ||Q 1_{||Q||_op > delta}||_HS^2 into {|y| <= R} and {|y| > R}:
   ||Q 1_{||Q||_op > delta}||_HS^2
     <= 2 ||D g_r 1_{||D g_r||_op > delta/2}||_2^2 + o(I) = o(I),   (8.35)
 
-after sending r -> 0 slowly along the sequence, then R -> infty, then
+after sending r -> 0 slowly along the sequence, then R -> infinity, then
 the sequence limit. The bound transfers to T_1 through
 ||T_1 - Q||_2 = o(sqrt I).
 
@@ -2377,7 +2377,7 @@ The eigenvalue exp(-t) of A contributes the exp(-2t) factor, the
 l-1 derivatives from T_l = (D^{l-1} T_1) contribute (2t)^{l-1},
 and projection is norm one on each chaos. Take the expansion to any
 fixed order m >= l-1, take the sequence limit at fixed m, then let
-m -> infty in the uniform remainder bound. The result is
+m -> infinity in the uniform remainder bound. The result is
 
   liminf((E|P(Y_0)_1 T_1(Y_t)|^2
     - c_{l,t} E|P(Y_0)_1 T_l(Y_0)|^2) / I) >= 0.   (8.37)
@@ -2464,7 +2464,7 @@ compactly supported P.
 For the stated P, membership in Dom(A_eta) puts P in H^2_loc by local
 elliptic regularity, and D P in L^2(eta). Choose smooth compactly
 supported cutoffs 0 <= chi_R <= 1 with chi_R -> 1 monotonically and
-||D chi_R||_infty -> 0. Equation (8.50) applies to chi_R P by local
+||D chi_R||_infinity -> 0. Equation (8.50) applies to chi_R P by local
 Sobolev approximation, and its right side is controlled by
 
   ||D(chi_R P) - DP||_2 -> 0,
@@ -2508,9 +2508,9 @@ corresponding flattening of T_{l+1}. This gives a clipped field C with
   ||C(y)||_op <= delta.   (8.66)
 
 Set P = e^{1-A} C = e P_1 C. Markov averaging and convexity of the
-operator-norm ball give ||P(y)||_op,infty <= e delta. The spectral
+operator-norm ball give ||P(y)||_op,infinity <= e delta. The spectral
 multipliers (x-1) e^{1-x}, e^{1-x}, x^{1/2} e^{1-x} are bounded on
-[0, infty); the function e^{1-x} - 1 vanishes at x = 1 and satisfies
+[0, infinity); the function e^{1-x} - 1 vanishes at x = 1 and satisfies
 |e^{1-x} - 1| <= C |x - 1| there. Combining these bounds with (8.5)
 and the derivative-closeness T_{l+1} = DF + o(sqrt I):
 
@@ -2518,7 +2518,7 @@ and the derivative-closeness T_{l+1} = DF + o(sqrt I):
 
   ||DP||_2 = O(sqrt I),   (8.68)
 
-  ||QP||_2 <= ||P||_op,infty ||Q||_2 <= e delta (1+o(1)) sqrt I.   (8.69)
+  ||QP||_2 <= ||P||_op,infinity ||Q||_2 <= e delta (1+o(1)) sqrt I.   (8.69)
 
 The clipping is used only to construct P, so it need not preserve
 tensor symmetry.
@@ -2562,7 +2562,7 @@ Together,
 
   <DF, K_s P>_2 = o(I).   (8.74)
 
-Replace P by DF. Since P - DF = o(sqrt I) and ||K_s||_op,infty <= s^{-1},
+Replace P by DF. Since P - DF = o(sqrt I) and ||K_s||_op,infinity <= s^{-1},
 
   |<DF, K_s (P - DF)>| <= ||DF||_2 s^{-1} ||P - DF||_2 = o(I).   (8.75)
 
@@ -2906,7 +2906,7 @@ Inserting f = 1, f = 1 + J, f = 1 - J and the identity
 All expressions are integrable: their absolute values are bounded by a
 constant multiple of tr(R^2 (1 + S)).
 
-Tail truncation. Let Q = 1_{(4, infty)}(S), P = 1 - Q. On this spectral
+Tail truncation. Let Q = 1_{(4, infinity)}(S), P = 1 - Q. On this spectral
 region (S - 1)^2/(1 + S) dominates both a positive constant and a
 positive multiple of S. Thus
 
@@ -3216,7 +3216,7 @@ approximately commute with R_*; hence each P_{m,t}(D) acts as a scalar
 matrix on block beta at leading order. The generating function of
 Hermite polynomials gives
 
-  sum_{j=0}^infty (2t)^{j/2}/j! h_j^{(1)}(x)
+  sum_{j=0}^infinity (2t)^{j/2}/j! h_j^{(1)}(x)
     = e^{x sqrt(2t) - t}.   (9.82)
 
 Therefore
@@ -3281,7 +3281,7 @@ above and below, and Gaussian entropy is continuous in its covariance.
 
 Proposition 31. There is an absolute constant C_0 such that, for
 every fixed t > 0 and L >= 1,
-  limsup ∫_0^infty MI(Y_t; Z_T) dT/T^2 <= C_0.   (10.6)
+  limsup ∫_0^infinity MI(Y_t; Z_T) dT/T^2 <= C_0.   (10.6)
 In particular C_0 = 9/4 suffices.
 
 Proof.
@@ -3338,6 +3338,6 @@ On a positive block beta with R eigenvalue beta,
   (h_T'(beta)/h_T(beta))^2
     = T beta^2 / ( (1 + T beta^2/L)^2 (1 + (1 + 1/L) T beta^2)^2 ).   (10.15)
 
-The right side has the required decay in beta: as beta -> infty, the
+The right side has the required decay in beta: as beta -> infinity, the
 denominator grows like (T beta^2/L)^2 (T beta^2)^2 while the numerator
 grows like T beta^2, so the ratio decays as beta^{-6}.