diff --git a/lattice_covering_n_log_n_v2.txt b/lattice_covering_n_log_n_v2.txt
index d61af08..ebde565 100644
--- a/lattice_covering_n_log_n_v2.txt
+++ b/lattice_covering_n_log_n_v2.txt
@@ -1589,21 +1589,13 @@ For an ordinary block the logarithm of each scalar alphabet size is
 O(D log(D+2) + b_i log b_i); the terminal version drops the second
 term. All implicit constants are absolute.
 
-In words, Proposition 10.1 packages the whole hierarchy into one
-object with four crucial features:
-
-  - RESIDUALS ARE SMALL: every retained point M ell is close to y in
-    every block, within the scale h_i of that block. This means the
-    Gaussian weight gamma_D(y - M ell) is at its natural magnitude,
-    not in some far tail.
-  - TOTAL WEIGHT IS A CONSTANT: sum_ell D_v gamma_D(y - M ell) >= kappa.
-    That fixed total is what supplies the horizontal load later.
-  - INDIVIDUAL WEIGHT IS TINY: each term is at most
-    exp(-c b_1^{0.70}), overwhelmingly small. That keeps each raw
-    slice fit for the common-lattice lemma.
-  - BINARY BY SUFFIXES: the pattern is a tree of binary choices, with
-    only the sibling separation +1. This is exactly the shape accepted
-    by the shear lemma of section 11.
+In words: (i) residuals are small in the h_i scale; (ii) total weight
+>= kappa > 0; (iii) each individual atom is <= exp(-c b_1^{0.70}), tiny;
+(iv) the pattern is binary by suffixes with sibling gap exactly 1.
+Properties (ii) and (iii) are complementary: the total supplies
+horizontal load, while the atom bound keeps each slice within reach of
+the mean-hole estimate. Property (iv) is exactly the shape the shear
+lemma of section 11 accepts.
 
 PROOF OF PROPOSITION 10.1.
 
@@ -1645,7 +1637,7 @@ attaining the maximum in (10.2), so g_1(x_1) = F_1(t_z)/p_1. By
 Lemma 9.4 there is lambda_1 in L_1 such that t = y_1/h_1 - lambda_1
 lies in t_z + [0, 1/p_1)^{b_1}. Consider all choices
 ell_1 = lambda_1 + eps for eps in {0,1}^{b_1}. Their individual block
-weights satisfy, by (9.19) with A-different eta_1 = delta_1,
+weights satisfy, by (9.19) with delta_1 in place of eta_1,
 
     (h_1^{b_1}/p_1) gamma_{b_1}( h_1 (t - eps) )
         >= e^{-delta_1} (F_1(t_z)/p_1) P_{1,t_z}(eps)
@@ -2077,24 +2069,12 @@ factor in density: (1 + 1/n)^n <= e.
 12. ASSEMBLING THE MAIN THEOREM
 ---------------------------------------------------------------------
 
-We now put everything together. The proof is a series of well-defined
-choices:
-
-  (1) Fix the absolute constants, especially the - extremely generous -
-      ultimate choice of C_* below. Fix the vertical hierarchy size.
-  (2) Get a finite list of horizontal slices from Lemma 7.1.
-  (3) Choose a horizontal covolume D_h so that each listed slice has
-      a small normalized volume u_a.
-  (4) Apply Lemma 7.2 to get ONE horizontal lattice Lambda_h whose
-      hole sets U_a on the torus are tiny simultaneously.
-  (5) Apply Lemma 11.1 to get one shear Z that, for every vertical
-      target and every retained pattern, makes the many hole sets
-      overlap only as if independent.
-  (6) Piece together: build the covering lattice Gamma with covolume
-      D_h D_v, prove its hole proportion is at most n^{-2n}, apply
-      Lemma 11.2 with k = n to close the last hole by dilating K by
-      the factor 1 + 1/n.
-  (7) Check the density is at most e C_* n log n.
+We now put everything together in ten steps. First fix the vertical
+hierarchy (Step 1); build a finite list of horizontal slices (Step 2);
+choose horizontal covolume and match loads (Steps 3-4); get one
+common horizontal lattice (Step 6); fix one shear (Step 7); build the
+covering lattice Gamma with tiny holes (Steps 8-9); and finally apply
+completion plus affine/small-dimension reductions (Steps 9-10).
 
 PROOF OF THEOREM (main).
 
@@ -2353,34 +2333,14 @@ dimensions is an absolute constant, absorbable into C. So the main
 theorem holds in the remaining dimensions too. That completes the
 proof of Theorem (main).
 
-A closing summary for the reader. The proof is completely self-contained
-after two imported theorems: the Eldan--Klartag marginal lower bound
-(EK) and the Rogers--Schmidt mean-hole estimate (RS). The Li--Liu binary
-correction is only a source of method; we re-prove it from scratch as
-Lemma 8.1. The logical chain is:
-
-    (EK) -> horizontal slices (Lemma 7.1)
-    folded Gaussian blocks (Lemma 9.1) -> usable anchors (Lemma 9.2)
-    -> branching hierarchy (Proposition 10.1)
-    (RS) -> one horizontal lattice with tiny holes (Lemma 7.2)
-    one simultaneous shear (Lemma 11.1) to make hole sets overlap
-    -> covering lattice Gamma with hole <= n^{-2n}
-    -> completion (Lemma 11.2) and dilation -> density O(n log n).
-
-Each step's input and output are explicit. The decimal exponents
-(0.56, 0.65, 0.69, 0.70, 23/32, ...) are only chosen to satisfy strict
-inequalities; they are not optimal. The constant C in the theorem is a
-finite number one could extract by carefully threading through these
-absolute constants; it is large, but absolute.
-
-What you should take from this proof, even without following every
-epsilon: the general shape is "cover most of each horizontal slice by
-one good lattice, stack enough vertical slices that the residuals
-cannot all line up, use one shear to make their holes behave
-independently, and dilate by a whisker to close the last leak." The
-surprise is that ONE lattice suffices; the work is getting the
-vertical pattern to have enough total weight while remaining uniformly
-thin.
+A closing summary. The logical chain is: (EK) -> horizontal slices
+(Lemma 7.1); folded Gaussian blocks (Lemma 9.1) plus usable anchors
+(Lemma 9.2) -> branching hierarchy (Prop 10.1); (RS) -> one horizontal
+lattice with tiny holes (Lemma 7.2); a simultaneous shear (Lemma 11.1)
+makes hole sets overlap as if independent; and finally completion
+(Lemma 11.2) plus dilation gives density O(n log n). The decimal
+exponents (0.56, 0.65, 0.69, 0.70, 23/32) are chosen only for strict
+inequalities, not optimality; the constant C is finite though large.
 
 ---------------------------------------------------------------------
 13. REFERENCES