diff b34550f..b1f9bc4 symplectic_ball_packings_v2.txt
diff --git a/symplectic_ball_packings_v2.txt b/symplectic_ball_packings_v2.txt
index eee0378..051ceb5 100644
--- a/symplectic_ball_packings_v2.txt
+++ b/symplectic_ball_packings_v2.txt
@@ -158,53 +158,6 @@ volume and all pairs are forced. The hard direction -- that
these two conditions are also enough -- is the rest of the file.
-4. HOW SUFFICIENCY IS ORGANIZED
-----------------------------------------------------------------
-Normalize R = 1 (Section 2.7). We must show: if
- R_1^n+...+R_k^n < 1 and R_i+R_j < 1 for all i<j,
-then the k balls pack into the open unit ball.
-
-4.1 One ball. If k = 1 the pairwise condition is empty and the
- volume condition says R_1 < 1, so the ball sits inside by
- inclusion. Done. Hence assume k >= 2 from now on.
-
-4.2 At most one big ball. A capacity is "big" if it is >= 1/2.
- If two were big, their sum would be >= 1, violating the
- pairwise condition. So there is at most one big ball. This
- splits the problem into exactly three cases:
-
- CASE 1. Every capacity is < 1/2. (Sec 9)
- CASE 2. One capacity A >= 1/2, and n >= 4. (Sec 10)
- CASE 3. One capacity A >= 1/2, and n = 3. (Sec 11)
-
- These cover everything. Section 12 assembles the three cases
- into the theorem.
-
-4.3 The engine common to all cases. Every case reduces to the
- same machine, which has three parts:
-
- PART A (Section 6). Turn a "the total leftover area is
- positive" statement (a single number, the mean of a function
- P over a polytope Delta) into a "the leftover area is
- positive at every point" statement, by sliding the fiber
- coordinates with a Hamiltonian map. Output: a strict
- pointwise gap.
-
- PART B (Section 7). Use a handle in a base surface to cut
- the model into disjoint "layers", one ball per layer.
-
- PART C (Section 8). Transport the abstract packing back into
- the actual ball, by normal-cone degenerations and Moser
- isotopies, while steering clear of forbidden divisors.
-
- Sections 9,10,11 then build, for each case, an explicit
- "base surface times toric fiber" model to which Parts A and B
- apply, and whose geometry Part C returns to the unit ball.
-
-We do these in order: Part A (Section 6), Part B (Section 7),
-Part C (Section 8), then the three cases.
-
-
5. HAMILTONIAN MAPS AND SORTING
----------------------------------------------------------------
Part A changes fiber coordinates with a Hamiltonian map. Here
@@ -1841,106 +1794,4 @@ The moral: once there are at least 3 complex dimensions, the extra
the two-ball rule.
-14. WHAT THE PROOF GIVES, AND WHAT IT DOES NOT
-----------------------------------------------------------------
-14.1 It is an EXISTENCE proof.
- It produces a packing when the two inequalities hold, and proves
- no packing exists otherwise. It does NOT give:
- - an algorithm to construct or draw the packing;
- - the shapes of the packed ball images (they are wildly bent);
- - effective numerical bounds on how small the neighborhoods
- around the closed source balls can be.
- That last point is why every "embedding" in this file means
- "symplectic embedding defined on an OPEN NEIGHBORHOOD of the
- closed source ball", and why the target is always the OPEN
- interior: no image may touch the boundary, and every image has
- a little room around it. Under finite composition of our
- constructions, the room is preserved (shrink neighborhoods at
- each step).
-
-14.2 The three explicit models, in one table.
- In all rows: target = unit ball (or shell), H(p) is the limiting
- base area, "cap" is (r_i - sum p_j)_+, and r_i > R_i are padded
- capacities. Each row's H has slopes whose drop keeps P concave.
-
- CASE 1 (every R_i < 1/2), Section 9.
- base C : complete intersection of n-1 quadrics in PP^n,
- deg 2^{n-1}, genus 1+(n-3)2^{n-2}, normal
- O_C(2)^{oplus(n-1)}.
- H(p) : 2^{n-1} (1 - 2 sum_{j=1}^{n-1} p_j).
- truncate : sum p_j <= h < 1/2.
- volume : J_m(h) = (2^{n-1} h^{n-1}/(n-1)!)(1 - 2(n-1)h/n)
- -> 1/n! as h -> 1/2.
- transfer : normal-cone (PP^n, C) -> PP^n minus H_inf -> ball.
-
- CASE 2 (one A >= 1/2, n >= 4), Section 10.
- base : C = quadrics curve inside hyperplane
- D = PP^{n-1} inside T = Bl_o PP^n; deg 2^{n-2},
- genus 1+(n-4)2^{n-3}.
- H(p) : 2^{n-2} (1 - p_1 - 2 w), w = sum_{j>=2} p_j.
- truncate : p_1 <= h_1 < s, w + h_2 p_1 <= h_2 < 1/2.
- volume : B_q(h_2)(1-(1-h_1)^n)/n -> (1-a^n)/n! (shell).
- transfer : cut/fiber chain -> PP^n minus (E union H_inf)
- -> shell; add central ball of capacity A < a.
-
- CASE 3 (one A >= 1/2, n = 3), Section 11.
- base : plane CUBIC C through two marks q_1,q_2,
- genus 1, deg_C O(1) = 3.
- H(p) : 9(b - p_1 - p_2) - 2(s - p_1 - p_2)_+,
- b = (1+s)/3, s = 1-a.
- truncate : p_1 <= h_1 < c < s, p_1 + p_2 <= h_2 near b.
- volume : int_{Delta_inf} H = (1-a^3)/6 (shell).
- transfer : moment-preserving transport -> Kähler cut ->
- normal-cone (Bl_o PP^3, quadric) -> shell;
- add central ball A < a.
-
- In every row the map P = H - sum_i cap_i is concave, has
- positive integral (that integral is a volume surplus), and is
- positive away from a central core; Part A (Section 6) upgrades
- this to a pointwise gap, Part B (Section 7) cuts disjoint
- layers, and Part C (Section 8) carries the packing home.
-
-14.3 What the proof uses from outside (the honest list).
- Everything is proved here except these named standard facts,
- each of which is a well-known theorem used as a black box:
- - Bertini's theorem and the Koszul/adjunction computations
- (Lemma 9.3): standard algebraic geometry.
- - the classification of line bundles on a bordered surface
- and on a genus-one curve (basepoint-freeness): standard.
- - the symplectic cut / Kähler quotient (Lemma 8.5): Lerman.
- - Moser's trick and its relative/equivariant refinement
- (Lemma 8.1, Lemma 7.7): standard.
- - the nested-disk rearrangement lemma (Lemma 6.10): standard
- planar area-preserving construction (Schlenk).
- - the invariant-cycle theorem for toric ampleness (Claim
- 11.17): standard toric geometry (Fulton et al.).
- - GROMOV'S TWO-BALL RIGIDITY (Section 3.2): the one
- genuinely symplectic input, via J-holomorphic spheres.
- With these imported, every step in between is written out.
-
-14.4 Where each piece lives (a reading map).
- statement + conventions .......... 1, 2
- necessity (volume, two-ball) ..... 3
- architecture, three cases ........ 4
- Hamiltonian flow, 1D sorting ..... 5
- Part A: comparison lemma ......... 6
- Part B: surface packing .......... 7
- Part C: Moser, normal-cone, bundle
- form, shell ............... 8
- Case 1 (all < 1/2) ............... 9
- Case 2 (big, n >= 4) ............. 10
- Case 3 (big, n = 3) .............. 11
- assembly = the theorem ........... 12
- dimension 4, caveats, summary .... 13, 14
-
-That is the complete proof, and the honest accounting of what it
-rests on. The two-line statement it establishes:
-
- for every n >= 3, every k >= 1, and all positive capacities,
-
- the balls pack into Int B^{2n}(R)
- iff sum_i R_i^n < R^n and R_i + R_j < R (i != j).
-================================================================
- end
-================================================================