diff --git a/markov_type_superreflexivity_v2.txt b/markov_type_superreflexivity_v2.txt
index 4be4e8f..26278d5 100644
--- a/markov_type_superreflexivity_v2.txt
+++ b/markov_type_superreflexivity_v2.txt
@@ -8,12 +8,10 @@ justified. Four standard facts are used from outside; they are listed in
 section 0.3, and the only hard one (James-Enflo) is used exactly once and
 flagged as a black box. Nothing else is assumed.
 
-HOW TO READ IT.
-  sections 0-3   the statement, the definitions, the plan
-  section 4      builds one special norm N on finite sequences
-  sections 7-10  builds one finite reversible random walk
-  section 5      where the hypothesis "Markov type p" is used
-  section 11     puts the pieces together and finishes
+HOW TO READ IT. Sections 0-3 are the statement, definitions and plan.
+Section 4 builds one special norm N on finite sequences; sections 7-10
+build one finite reversible random walk; section 5 is where the hypothesis
+"Markov type p" is used; section 11 puts the pieces together.
 
 0. THE THEOREM
 
@@ -53,13 +51,11 @@ set A, 1_A is its indicator function, equal to 1 on A and 0 off A.
       colours has a monochromatic subset of size s. Used in 4.3 and 7.2,
       always applied a fixed finite number of times in a row.
 
- (P3) Goldstine's theorem. For a Banach space Y with unit ball B_Y, the
-      weak-star closure of B_Y inside Y** is the unit ball of Y**. We
-      use only this consequence: for every v in the unit ball of Y**,
-      every finite list g_1,...,g_s in Y* and every delta > 0 there is y
-      in B_Y with |g_h(y) - v(g_h)| <= delta for h = 1,...,s. A short
-      proof of exactly this form is included in 4.2, so nothing is
-      hidden.
+ (P3) Goldstine's theorem, in the one form used: the unit ball of Y is
+      weak-star dense in the unit ball of Y**, that is, for every v in
+      the unit ball of Y**, every finite list g_1,...,g_s in Y* and
+      every delta > 0 there is y in B_Y with |g_h(y) - v(g_h)| <= delta
+      for each h. Proved in 4.2, so nothing is hidden.
 
  (P4) James-Enflo characterisation (restated in 2.1): X is superreflexive
       if and only if every Banach space finitely representable in X is
@@ -360,9 +356,12 @@ subadditive, and it is positive definite by (4.6).
 
 THREE PROPERTIES.
 
-(a) BOUNDS. Pass to the limit in (4.3) and (4.4) along the subsequence,
-noting that M = M_{k_j} -> inf. For every array a with prefix sums
-A_j = a_1 + ... + a_j,
+(a) BOUNDS. By (4.5) any placement of the coefficients among the units
+gives the same number in the limit, so we may compute N_0 with a_1 on the
+smallest chosen unit, a_2 on the next, and so on. In that reading (4.3)
+applies directly: the suffix sums of a reversed list are the prefix sums
+of a. Letting M = M_{k_j} -> inf, and using (4.4) as well, gives for every
+array a with prefix sums A_j = a_1 + ... + a_j
 
      d max_j |A_j| <= N_0(a) <= sum_i |a_i|.                           (4.6)
 
@@ -1439,26 +1438,18 @@ Theorem 0.1 supplies the missing direction. Naor's question -- must
 nontrivial Markov type force an equivalent uniformly smooth norm? -- has
 answer yes.
 
-12.4 Why Rademacher type is not enough. Rademacher type only tests sums
-with independent random signs. James's example of a nonreflexive space of
-Rademacher type 2 shows that such sums cannot see nonreflexivity. Markov
-type tests all finite reversible walks, and section 10 shows how a walk
-can be arranged so that consecutive summation converts a long reduced word
-into a linear displacement. That is the whole difference between the two
-notions, and it is a genuine one.
-
-12.5 The argument in one paragraph. Suppose X has Markov type p > 1. If X
-were not superreflexive, then by James-Enflo a nonreflexive space Y sits
-finitely representably inside X, and section 4 turns that into a norm N on
-finite sequences whose finite pieces still fit inside X almost isometrically,
-with two extra properties: inserting zeros does not change N, and summing
-adjacent entries does not increase N. Sections 7-10 then build, for every
-n, a finite reversible marked chain on charts whose edge labels all have
-N-norm 1 and whose n-step sum has N-norm at least n/24 with probability at
-least 3/20; the labels are transported along the chain by the two-level
-cocycle of section 9, and the prefix sums of the total label are forced to
-alternate between the two levels, which consecutive summation converts into
-a norm lower bound linear in the reduced length of the random word. Markov
-type, applied through the lattice lift of section 5, bounds the p-th moment
-of that sum by K^p n. But the walk shows it is at least a constant times
-n^p. Since p > 1, that fails for large n, and X must be superreflexive.
+12.4 Why Rademacher type is not enough. Rademacher type tests only sums
+with independent random signs, and James's nonreflexive space of Rademacher
+type 2 shows that such sums cannot see nonreflexivity. Markov type tests
+every finite reversible walk, and section 10 arranges a walk whose
+consecutive summation turns a long reduced word into a linear displacement.
+
+12.5 The argument in one paragraph. If X were not superreflexive, then by
+James-Enflo a nonreflexive space Y sits finitely representably inside X;
+section 4 turns that into a norm N on finite sequences whose finite pieces
+still fit inside X almost isometrically, with zeros insertable for free and
+adjacent summation nonexpansive; sections 7-10 give, for every n, a finite
+reversible walk whose labels all have N-norm 1 and whose n-step sum has
+N-norm at least n/24 with probability at least 3/20; section 5 turns that
+into E N(sum)^p <= K^p n; and (3/20)(n/24)^p <= K^p n is impossible for
+large n once p > 1.