diff --git a/qma_coulomb_hardness_v2.txt b/qma_coulomb_hardness_v2.txt
index 7c91e95..c1e7ef4 100644
--- a/qma_coulomb_hardness_v2.txt
+++ b/qma_coulomb_hardness_v2.txt
@@ -22,6 +22,16 @@ CONTENTS
  12. Implementability and encoding
  13. Putting the proof together
 
+READING GUIDE
+  ESSENTIAL: Sections 1-4 (what the theorem says, what a molecule is, the
+  promise-problem format, the one-page plan) and Section 13 (how the proof
+  ties together). Together about 8 pages. Read this path first.
+  THE PROOF: Sections 5-11 build the molecule. Each section takes the
+  previous one as input, adds a gadget, and produces a stronger object.
+  TECHNICAL: Section 12 checks that every output is computable in
+  polynomial time. Skippable on a first read; nothing later needs it.
+  FIRST-READ SKIPPABLE: 6.1, 8.2, 10.2 (technical estimates).
+
 CONVENTIONS
   Atomic units throughout. |A| = volume of a set A. "p(n,D)" means some
   polynomial in the named variables whose coefficients and exponents are
@@ -817,6 +827,14 @@ U - v_{ij} below is uniformly positive and bounded.
 8.3 The occupancy penalty and superexchange
 --------------------------------------------------------------------------------
 
+IN ONE BREATH. Two electrons on neighbouring sites with opposite spins
+can momentarily pile onto a single site and then split back. Pauli's
+principle permits the pile-up only when the two spins are antiparallel
+(a singlet); for parallel (triplet) spins the first hop is blocked. So
+the double hop lowers the singlet's energy relative to the triplet's by
+roughly t^2/(U - v), i.e. a favourable spin exchange whose size is set
+by the virtual-excitation cost.
+
 THE FERMION MODEL. Consider 2m canonical fermion modes: at each site i, two
 spin states sigma in {up,down}. Restrict to exactly m electrons (the
 filling). Let n_i = sum_sigma c_{i sigma}^dagger c_{i sigma} be the total
@@ -1398,6 +1416,14 @@ errors are O(lambda^{-1/3}) in the units of H_d, up to polynomial factors.
 10.1 Transport and charge placement
 --------------------------------------------------------------------------------
 
+IN ONE BREATH. To replace a continuous charge smear by point nuclei,
+tile the uniform slab it was built from by an evenly spaced cubature and
+let a smooth vector field push each cubature node toward the thicker
+region. Choose the flow so the pushed nodes' local density equals the
+target smear everywhere; then the discrete Coulomb field mimics the
+continuum field on all smooth test functions, and every node keeps its
+common weight (charge one). This is Moser's construction.
+
 The density interpolation is Moser's deformation method (Moser 1965;
 Dacorogna-Moser 1990). Here the vector field is EXPLICIT and compactly
 supported inside the box, and we prove the required transport directly from
@@ -1655,6 +1681,14 @@ m-electron space of 2m one-particle spin modes.
 11.1 Control outside the orbital space
 --------------------------------------------------------------------------------
 
+IN ONE BREATH. We want a lower bound on the FULL ground energy, but we
+only know the modes well. Split any trial state as w (modes) plus q (the
+rest). The rest costs at least alpha times its H^1 norm; the modes part
+costs at least E_Pi; the only danger is the cross-coupling <q, Hcal w>,
+bounded by beta times ||q|| times ||w||. Completing the square in ||q||:
+alpha ||q||^2 - 2 beta ||w|| ||q|| >= -beta^2/alpha * ||w||^2. Hence
+E_full >= E_Pi - beta^2/alpha, up to polynomial lambda factors.
+
 We first give the COMPLEMENT a uniform lower bound and bound its coupling to
 the mode space. Then a square completion bounds the FULL infimum.