diff --git a/qma_coulomb_hardness_v2.txt b/qma_coulomb_hardness_v2.txt
index 118471a..3e8ebd0 100644
--- a/qma_coulomb_hardness_v2.txt
+++ b/qma_coulomb_hardness_v2.txt
@@ -180,7 +180,7 @@ which every fresh pair is a singlet; on that space the original signed
 terms and the source-unaffected part act unchanged. The kernel restriction
 of the first-order term P B P keeps the unreplaced part. The second-order
 correction is -P B A^{-1} B P. Only two "hops" through B can act on the
-same fresh pair, and they have energy denominator g = 4 Delta (the gap of
+same fresh pair, with energy denominator g = 4 Delta (the gap of
 A on that pair): so A^{-1} on a double action on pair e is 1/(4 Delta)
 times the pair identity.
 
@@ -252,8 +252,7 @@ gadgets and the subdivisions and their second-order corrections) with
 
 Every intermediate weight K_e and its shift are computable (as real
 numbers) to inverse-polynomial absolute accuracy in polynomial time.
-Square roots need not be exact rationals at this stage; they will be
-absorbed into a final rational approximation.
+Square roots need not be exact rationals at this stage; a final rational approximation absorbs them.
 
 The positive-exchange Heisenberg Hamiltonian H_+ = sum K_e h^s_{ij} has polynomial size m, every K_e and 1/K_e polynomially bounded, and a scalar shift C_s satisfying (9).
 
@@ -455,7 +454,7 @@ Define the direct electron-electron coefficients
     U = v(0) .
 
 (U is the same-site coefficient; v_{ij} is the coefficient between sites i,j.)
-They are uniformly bounded (the psi_i are uniformly bounded and decay), and
+These are uniformly bounded (the psi_i are uniformly bounded and decay), and
 v is uniformly Lipschitz (translate the decaying density and differentiate).
 
 Coercivity (a fixed lower bound on the whole matrix v). Now prove
@@ -707,7 +706,7 @@ counterterm, formula (30); a fixed one-particle gap outside the modes
 
 Everything is in dilated coordinates y = (r,z); energies are the dilated
 ones (divide final energies by lambda^2 later). The nuclei are not yet
-points; they will be the density's field.
+points; these will be the density's field.
 
 Choose integers
 
@@ -724,7 +723,7 @@ box; because the box is tall and wide, near the planar wells F_H behaves
 like a constant minus a vertical harmonic well centered on z=0, plus small
 error (see (34)).
 
-Why we need a well-depth adjustment. The occupancy identity (C) of the
+Why well depths need adjusting. The occupancy identity (C) of the
 superexchange calculation requires a one-body diagonal term -s_i/lambda on
 site i, where
 s_i = sum_{j!=i} v_{ij}. In the mode basis, the diagonal entry from
@@ -742,9 +741,9 @@ put
     b_i = s_i / ( - int varphi^2 W ) >= 0 ,
     V_ell(r,z) = zeta(z/S) sum_i (1 + b_i/lambda) W_i(r) .       (35)
 
-The b_i are nonnegative and polynomially bounded independently of D (they
-depend only on the geometry of the wells through v_{ij} and the fixed
-integral -int varphi^2 W). Hence 0 <= b_i/lambda < 1 at the scales. The
+The b_i are nonnegative and polynomially bounded independently of D,
+depending only on the geometry of the wells through v_{ij} and the fixed
+integral -int varphi^2 W. Hence 0 <= b_i/lambda < 1 at the scales. The
 functions V_ell are supported in disjoint vertical cylinders around the u_i,
 inside Omega.
 
@@ -835,7 +834,7 @@ orthonormal modes
     || Gamma^{-1/2} - I || <= p(n,D) lambda^{-0.8} .                (39)
 
 Then unit vectors in Ran P have polynomially bounded one-particle kinetic
-energy (they are finite combinations of the smooth modes with weights of
+energy (finite combinations of the smooth modes with weights of
 polynomial size).
 
 Proposition 1 (continuous one-particle control).
@@ -917,7 +916,7 @@ since e_* = -1/2 + omega/2 makes the constant term zero. Choosing the
 polynomial scale so the total error O(S^3/H) + O(p(n,D)/lambda) + o(1) is
 at most c/2 gives the first inequality in (40). The slowly small errors
 o(d^{-2}), o(s^{-2}) are only needed to establish a fixed gap, not a per-
-particle energy accuracy, so they do not blow any budget.
+particle energy accuracy; no budget is blown.
 
 Step 5: kinetic lower bound and the mode-space norm. The kinetic lower bound
 L_c >= -Delta_y/2 - C follows from (34) (F_H - C_H bounded below by
@@ -958,7 +957,7 @@ lambda^{-1} sum_k b_k W_k on psi_j is lambda^{-1} b_j I = lambda^{-1}
 ( s_j / (-I) ) I = - s_j / lambda. Since s_j = sum_{i != j} v_{ij}, this is
 the same number as -lambda^{-1} sum_{i != j} v_{ij} promised in the
 construction. The other-well terms sum_{k != j} W_k acting on psi_j carry two
-planar tail factors (spacing >= 0.95 D) and are exponentially small, so they
+planar tail factors (spacing >= 0.95 D) and are exponentially small, so these
 do not affect the diagonal. Hence A_jj = -s_j/lambda up to negligible error.
 
 Row i != j (off-diagonal). In the first sum sum_{k != j} W_k, the term with
@@ -999,7 +998,7 @@ with H_d of (20), whose hopping is -t_e on links and whose penalty
 produces -s_i as the linear counterterm, one sees the compressed one-particle
 matrix is exactly the one-particle part of H_d, up to p(n,D) lambda^{-0.1}.
 
-Where we are. There is a continuous charge density whose one-particle
+So far. There is a continuous charge density whose one-particle
 operator L_c has (i) the modes as approximate low-energy states with a fixed
 complementary gap, and (ii) a compressed matrix equal to the one-particle
 part of H_d to accuracy p(n,D) lambda^{-0.1}. The cubature replacement replaces the
@@ -1167,7 +1166,7 @@ Mathematics 14, AMS, 2001. ISBN: 978-0821827833.
   order at least two together with a kernel derivative bounded by
   C|y - G(b)|^{-j}, 2 <= j <= 4. Such terms are nonzero only on cells meeting
   the well cylinders (combined slab volume O(m(S+1))). Below distance one
-  they are bounded by the same inverse fifth power; beyond distance one their
+  these are bounded by the same inverse fifth power; beyond distance one their
   total contribution is at most C h^4 m(S+1).
 
 So the pointwise absolute error is at most C(h^2 + h^4 m(S+1)) plus the
@@ -1264,7 +1263,7 @@ continuity from the core. No ground eigenvector is assumed.
 The projection Pi. Let Pi project onto the subspace in which all m particles
 lie in P (the m-electron space built from the 2m one-particle spin modes).
 Write ||u||_1^2 = ||u||^2 + sum_l ||nabla_l u||^2. Because the modes are
-smooth and decaying, they lie in the form domain, so the compression
+smooth and decaying, these lie in the form domain, so the compression
 (Pi Hcal Pi)|_{Ran Pi} is well defined and equals, by (51), the finite
 fermion operator below. The one-particle kinetic bound (39) gives
 ||w||_1 <= p(n,D) ||w|| for w in Ran Pi.
@@ -1469,7 +1468,7 @@ and the reduction meets the required output separation b - a >= 1.
 
 The configuration and thresholds are next certified to be computable in deterministic polynomial time with polynomial encoding length. The scale order is: (1) bounded well data (f, phi, W) and fixed integer rho from derivative bound (30); (2) source cutoff and spin-gadget penalties Delta, Delta', Delta'' from n and gamma by the normalization to rationals; (3) rational tau from the uniform Coulomb coercivity constant c and the polynomial size bounds; (4) integers k = ceil(C n^A), h = 1/k, lambda = 8 k^3/rho with A, C fixed absolute positive integers then large enough.
 
-At stage 4, K_e, 1/K_e, the layout extent, and the b_i are already polynomial in n, so they do not affect k. Every remaining condition is p(n, log lambda) lambda^{-alpha} <= inverse polynomial in n, and the hopping endpoint brackets of (27) sit at the same fixed-power rate; the polynomials p(n,D) come from fixed-order derivatives and fixed-dimensional data, and the large box enters only through (34) and summable cubature derivatives. A large enough A meets every asymptotic requirement; C covers all n >= 2 plus the fixed conditions; hence H, |Omega|, and 1/delta_R are polynomial in n.
+At stage 4, K_e, 1/K_e, the layout extent, and the b_i are already polynomial in n, independent of k. Every remaining condition is p(n, log lambda) lambda^{-alpha} <= inverse polynomial in n, and the hopping endpoint brackets of (27) sit at the same fixed-power rate; the polynomials p(n,D) come from fixed-order derivatives and fixed-dimensional data, and the large box enters only through (34) and summable cubature derivatives. A large enough A meets every asymptotic requirement; C covers all n >= 2 plus the fixed conditions; hence H, |Omega|, and 1/delta_R are polynomial in n.
 
 The large scalar shift. The dominating scalar is C_H times lambda^2 m; evaluating it within
 
@@ -1541,7 +1540,7 @@ reference density or any cell mass.
 
 None of these operations uses a many-electron wavefunction: only single-particle quadrature, fixed-dimensional flow integration, and rational rounding. There are 8 |Omega| h^{-3} point nuclei, polynomially many in n, each of charge exactly one. Their rational physical positions x~_alpha = b~_alpha / lambda, the electron number N = m (in unary), and the rational thresholds a, b of (84) all have polynomial encoding length and are computable by the scale order, numerical evaluation, and flow evaluation. Reducing rational outputs to lowest terms gives the input format of the promise problem.
 
-On any input, first verify source syntax and the fixed polynomial bounds used in the construction; if they fail, output a fixed instance (which obeys its own promise vacuously and does not affect either promised implication). Otherwise run the deterministic construction above.
+On any input, first verify source syntax and the fixed polynomial bounds used in the construction; on failure, output a fixed instance (which obeys its own promise vacuously and does not affect either promised implication). Otherwise run the deterministic construction above.
 
 The affine map F is (82). The estimates (9), (29), (78) and (81) chain to a single bound on the source-instance molecule,