diff 68ff689..0e5a8de symplectic_ball_packings_v2.txt
diff --git a/symplectic_ball_packings_v2.txt b/symplectic_ball_packings_v2.txt
index e69de29..c4b316f 100644
--- a/symplectic_ball_packings_v2.txt
+++ b/symplectic_ball_packings_v2.txt
@@ -0,0 +1,1481 @@
+================================================================
+================================================================
+ SYMPLECTIC BALL PACKINGS IN HIGHER DIMENSIONS
+ a reader's guide
+================================================================
+================================================================
+
+A friendly tour for a high-school reader who knows some
+calculus and has met complex numbers, but has never seen
+symplectic geometry. We don't reprove the technical lemmas
+here. Our only goal is to explain what the paper proves, why
+it is hard, and how the pieces fit together, so a curious
+young reader can follow the actual paper afterwards.
+
+The technical paper has the same title (September 2026); this
+file is the guide to it.
+
+
+1. THE ONE-PARAGRAPH ANSWER
+----------------------------------------------------------------
+
+Imagine a big "symplectic ball" of capacity R (we will define
+capacity in a moment) and k small closed symplectic balls of
+capacities R_1, ..., R_k. Can the small balls be fit disjointly
+into the interior of the big one by a symplectic map?
+
+Dimension is important. We work in n complex dimensions,
+which is 2n real dimensions. The theorem covers every n >= 3,
+every k >= 1, and every choice of positive capacities.
+
+Main theorem (this paper). The packing is possible
+IF AND ONLY IF both of the following hold:
+
+ R_1^n + R_2^n + ... + R_k^n < R^n (volume)
+
+ R_i + R_j < R for every pair i != j (two-ball)
+
+There are no other conditions. Not one extra constraint.
+Not in dimension 6, not in dimension 8, not ever.
+
+That is exactly Siegel and Yao's Conjecture A, stated in 2025 and
+now proven positively in every n >= 3.
+
+The strict inequalities reflect the "closed sources, open target"
+convention: the closed balls must live inside the open interior
+of the big ball, so no volume can leak onto the boundary.
+
+
+2. WHAT IS A SYMPLECTIC BALL
+----------------------------------------------------------------
+
+We work in C^n, which as a real vector space is just R^{2n}.
+Write a point as z = (z_1, ..., z_n) with z_j = x_j + i y_j.
+
+The closed symplectic ball of capacity R is
+
+ B^{2n}(R) = { z in C^n : pi * ( |z_1|^2 + ... + |z_n|^2 ) <= R }
+
+where |z_j|^2 = x_j^2 + y_j^2. This is a round ball of
+Euclidean radius sqrt(R/pi). Its volume is R^n / n!.
+
+We will always measure balls by capacity R, never by radius.
+Why? Because symplectic geometry sees "symplectic area," and
+the symplectic area of a complex 2-plane through the origin is
+exactly pi * r^2 = R. Capacity is the natural unit.
+
+The symplectic form on C^n is
+
+ omega_0 = dx_1 ^ dy_1 + dx_2 ^ dy_2 + ... + dx_n ^ dy_n.
+
+Each term dx_j ^ dy_j is the signed area element of the
+j-th complex coordinate plane C_j = { other coords = 0 }.
+So omega_0 is a sum of n independent signed area measurements.
+
+A symplectic embedding f : U -> U' (both open subsets of C^n)
+is a smooth injective map with f^* omega_0 = omega_0.
+That is: the pullback of the area form equals the original.
+
+Volume of a region is the integral of the top power
+omega_0^n / n!. So preserving omega_0 automatically preserves
+volume. But preserving omega_0 is much stricter than preserving
+volume, because omega_0 records signed 2-dimensional area on
+every coordinate plane, not only the total 2n-dimensional
+volume.
+
+For n = 1 (ordinary plane), preserving omega_0 = dx ^ dy is
+the same as preserving ordinary area. There is no rigidity.
+For n >= 2 a gap opens between "symplectic" and
+"volume-preserving," and that gap is the whole story.
+
+Gromov's fundamental observation: symplectic geometry is
+rigid in a way that volume-preserving geometry is not, in
+every dimension n >= 2.
+
+
+3. THE TWO OBSTRUCTIONS
+----------------------------------------------------------------
+
+Both parts of the main theorem's necessity are short to state.
+
+(3a) VOLUME
+ Symplectic maps preserve volume. The volume of B^{2n}(R) is
+ R^n / n!, so the total volume of the packed balls must not
+ exceed the target volume:
+
+ (R_1^n + ... + R_k^n) / n! < R^n / n!
+
+ which is exactly the first inequality.
+
+(3b) TWO-BALL (Gromov, 1985)
+ If you have two balls of capacities R_1, R_2 that you want
+ to pack disjointly into capacity R, then
+
+ R_1 + R_2 < R.
+
+ This is the surprising one. Two half-capacity balls in
+ dimension n = 3 have total volume just 1/4 of the target,
+ yet they cannot fit: their capacities add to 1, so any
+ symplectic map that tries to place both is forbidden.
+
+ The proof idea, in one paragraph:
+
+ A symplectic form like omega_0 admits a compatible
+ almost-complex structure J (a rule for rotating tangent
+ vectors by 90 degrees, like multiplication by i in
+ coordinates). Once you have J, you can ask for
+ J-holomorphic spheres: maps from the standard round
+ two-sphere S^2 into the symplectic manifold whose
+ derivative is complex-linear.
+
+ Gromov proved that through every point of C^n there
+ passes such a sphere, and its symplectic area is
+ pinned to a specific value tied to the "size" of the
+ ball it is drawn through. Those spheres behave like
+ impossible-to-shrink rulers.
+
+ Now, if you try to place a second ball B' disjointly
+ from a first ball B, the Gromov sphere through B
+ cannot touch B'. That forces B' to be below a
+ calculable capacity. Doing this carefully gives
+ R_1 + R_2 < R.
+
+ A full proof uses the moduli space of J-holomorphic spheres
+ and their intersection theory. That is far beyond high
+ school. The takeaway that matters: these rigid curves
+ capture information about size that ordinary volume does
+ not see.
+
+(3c) NECESSITY OF THE SECOND OBSTRUCTION FOR ALL PAIRS
+ If you have k >= 3 balls and any pair, say i and j, has
+ R_i + R_j >= R, then those two alone cannot fit. Since
+ the whole packing contains that pair, the whole packing
+ cannot fit either. Applying this over all pairs gives
+ the full set of R_i + R_j < R conditions.
+
+So the necessity direction of the theorem is short.
+
+
+4. THE SUFFICIENCY STRATEGY IN ONE PAGE
+----------------------------------------------------------------
+
+Normalize the target capacity to 1. A single scaling factor of
+sqrt(R_0/R) applied to both source and target gives this for
+free; the two factors cancel inside the pullback.
+
+If there is only one ball, place it inside the larger target by
+inclusion: done.
+
+Otherwise, the theorem splits into three geometric cases:
+
+CASE 1: EVERY REQUESTED CAPACITY IS < 1/2.
+ Build a "positive-genus curve" model inside PP^n and pack
+ everything there. The curve is the complete intersection of
+ n-1 smooth quadric hypersurfaces. Its genus is at least 1
+ for every n >= 3, so we get the "handle" required by the
+ surface packing lemma.
+
+CASE 2: EXACTLY ONE CAPACITY, SAY R_0, IS >= 1/2, AND n >= 4.
+ Reserve a ball of capacity a > R_0 at the center. Pack the
+ remaining balls into the shell { a < sum |z_j|^2 < 1 }.
+ For n >= 4, a two-stage model works: first a hyperplane
+ inside PP^n, then inside that hyperplane a complete
+ intersection of quadrics. The second-stage curve has
+ genus 1 + (n-4) 2^{n-3}, which is >= 1 for n >= 4.
+
+CASE 3: EXACTLY ONE CAPACITY, SAY R_0, IS >= 1/2, AND n = 3.
+ Same reservation idea, but for n = 3 the plane inside PP^3
+ is PP^2, and a conic in PP^2 has genus 0 (no handle). We
+ replace the conic by a plane cubic through two marked
+ points. Its genus is 1 and removing the marks keeps a
+ puncture. A further "Kähler cut" makes the shell available.
+
+In Case 2 or 3, "one capacity >= 1/2" is automatic uniqueness:
+since every pair satisfies R_i + R_j < 1, another ball with
+capacity >= 1/2 would force R_i + R_j >= 1, contradiction.
+
+The engine behind every case is the following three-part
+machine, in order:
+
+ STEP A: turn volume into pointwise room
+ (Sections 5 of this guide, corresponding to comparison.tex
+ of the paper). On a polytope Delta, a continuous concave
+ function P with positive mean is upgraded to a strict
+ pointwise inequality via a Hamiltonian isotopy.
+
+ STEP B: turn pointwise room into layers
+ (Section 6, corresponding to surface.tex). A handle in the
+ base surface provides a closed 1-form with period 1, which
+ creates a stack of disjoint "layers" along a coordinate.
+ One closed ball goes into each layer.
+
+ STEP C: transport into the real target
+ (Section 7, corresponding to geometry.tex). Normal-cone
+ degenerations and Moser isotopies move the packing from the
+ abstract model into the actual ball, while preserving
+ disjointness and keeping away from forbidden divisors.
+
+Case-specific details (Sections 8 of this guide) show how to
+produce an appropriate "surface plus toric fiber" model whose
+volume is large enough for each of the three cases.
+
+
+5. STEP A: HAMILTONIAN COMPARISON
+----------------------------------------------------------------
+
+This is the beating heart of the proof and lives in
+comparison.tex. We take a mean inequality and upgrade it to a
+strict pointwise inequality by reshuffling the fiber coordinates.
+
+SETTING.
+ Fix m >= 1 and a compact polytope
+
+ Delta = { p in R^m_{>=0} : b_nu . p <= c_nu, nu = 1..N }
+
+ with b_nu in R^m_{>=0} (nonnegative coefficients) and
+ positive bounds c_nu. If b_nu . p = c_nu we call it an
+ "outer face" of Delta. The coordinate faces p_j = 0 are
+ NOT outer faces.
+
+ In C^m, take the toric domain
+
+ V = { z in C^m : p(z) in Delta }, p_j(z) = pi |z_j|^2,
+
+ with the toric form
+
+ omega_V = dp_1 ^ d theta_1 + ... + dp_m ^ d theta_m
+
+ where theta_j = arg(z_j) / (2 pi). This is just the standard
+ symplectic form on C^m, restricted to V.
+
+Think of Delta as the "moment polytope" of V: it records how much
+area is available along each complex coordinate direction.
+Lowering any moment p_j keeps you in Delta (because b_nu has
+nonnegative coefficients), so Delta "opens downward."
+
+THE LEMMA.
+ Let P : Delta -> R be continuous and concave. Suppose
+
+ (i) the average of P over Delta is strictly positive;
+ (ii) there exists 0 < tau < 1 such that
+ P is bounded below by a positive constant on
+ Delta minus tau Delta (i.e. outside a shrunken
+ copy of Delta near its center).
+
+ Then there is a smooth toric function K on Delta and a
+ Hamiltonian diffeomorphism q : V -> V with
+
+ ( K - P )( q u ) < K( u ) for every u in V.
+
+ The Morse-theoretic way to say this: the isotopy from the
+ identity to q is generated by Hamiltonians that Poisson
+ commute with the defining moment b_nu . p near each outer
+ face b_nu . p = c_nu, uniformly in time.
+
+WHY THIS HELPS.
+ Later, when we build the "layers," we need a strict
+ pointwise gap between two functions. The lemma produces
+ exactly that gap from a mean-only hypothesis. Hypothesis
+ (ii) prevents the leftover area from vanishing inside the
+ polytope in a way that defeats the argument.
+
+PROOF, IN THREE MOVES.
+
+ MOVE 1: increasing rearrangement.
+
+ Given a continuous convex f : [0,1] -> R, define its
+ increasing rearrangement g by
+
+ g(h) = min_{0 <= a <= 1-h} max{ f(a), f(a+h) }.
+
+ The name is literal. The rearranged function is the "sorted"
+ version of f, made nondecreasing while preserving:
+ - the integral of f over [0,1];
+ - order (f <= g implies R_f <= R_g);
+ - translation by a constant;
+ - uniform norm (|R_f - R_g| <= |f - g|).
+
+ For a function F(y, p_j) on Delta, define R_j F by taking
+ the slice at fixed other moments y, viewing F as a convex
+ function of p_j on [0, L(y)] with L(y) the largest p_j allowed
+ by Delta, and applying the 1D rearrangement scaled to the
+ slice length.
+
+ Key property: R_j preserves convexity. Reason: on two
+ slices of lengths h_0 and h_1, if we take the convex
+ combination (1-t) h_0 + t h_1 and the corresponding convex
+ combination of minimizing intervals, F evaluated at the two
+ endpoints of that interval is bounded by the same convex
+ combo of F at the two original endpoints, so the max-formula
+ inherited by the rearrangement stays convex.
+
+ Apply R_1, R_2, ..., R_m in order; call the composite R.
+ Since each R_j is nondecreasing in p_j and preserves
+ monotonicity in other coordinates, R F is coordinatewise
+ nondecreasing.
+
+ MOVE 2: contraction fixed point.
+
+ For 0 < lambda < 1, construct
+
+ F |--> R( lambda F - P ).
+
+ The map is a contraction of the complete metric space of
+ continuous convex functions on Delta, in uniform norm, with
+ constant lambda. So it has a unique fixed point
+ K_lambda.
+
+ Because R preserves integrals (Move 1) and P is fixed:
+
+ mean( K_lambda ) = -mean(P) / (1 - lambda).
+
+ In particular, mean(K_lambda) -> -infinity as lambda -> 1.
+
+ Now choose a small positive d < min( mean(P),
+ inf over
+ Delta minus tau Delta
+ of P ),
+
+ strictly less than both the mean of P and the boundary
+ lower bound of P.
+
+ Claim: for lambda close enough to 1,
+
+ M_lambda := max_over_Delta K_lambda < - d / (1 - lambda).
+
+ Prove by contradiction. If M_lambda is not that negative,
+ then on the outer region Delta minus tau Delta we can use
+ lambda K_lambda - P <= M_lambda + d (small),
+ while on the inner core tau Delta we can use convexity to
+ bound K_lambda far below M_lambda, giving
+ lambda K_lambda - P << M_lambda - something big.
+ The "something big" blows up as lambda -> 1, so on both
+ regions we end up with
+ max( lambda K_lambda - P ) < M_lambda,
+ contradicting the fixed-point equation
+ K_lambda = R( lambda K_lambda - P )
+ and the fact that max( R F ) <= max( F ).
+
+ So the claim holds for lambda close enough to 1.
+
+ Fix such a lambda. The pointwise inequality
+ (1 - lambda) K_lambda <= - d
+ (which is exactly M_lambda < -d/(1-lambda) rearranged) plus
+ R's order-preservation and translation-invariance gives
+
+ R( K_lambda - P ) <= K_lambda - d.
+
+ So: if we apply the full rearrangement R to F_0 := K-P,
+ the output is at most K - d. Strict gap of size d.
+
+ MOVE 3: realize rearrangement by Hamiltonian maps.
+
+ The rearrangement R = R_m ... R_1 is abstract. The paper
+ proves it can be implemented, up to arbitrarily small error,
+ by an actual Hamiltonian diffeomorphism of V that permutes
+ only the fiber variables.
+
+ The key sub-lemma is 2D:
+
+ Planar rearrangement lemma. Let f_y : [0,1] -> R be a
+ jointly continuous family of convex functions,
+ g_y the increasing rearrangements. For every eps > 0,
+ there is a smooth family of compactly supported
+ Hamiltonian isotopies of the disk D(1), with endpoints
+ psi_y, such that for every z in D(1),
+
+ f_y( pi |psi_y(z)|^2 ) <= g_y( pi |z|^2 ) + eps.
+
+ The idea: on D(1), the "moment" pi |z|^2 is the area of
+ the concentric disk through z. So we want a Hamiltonian
+ map that "sorts" the values of f into increasing order.
+ The trick is:
+ - choose a fine grid 0 < h_1 < ... < h_l < 1;
+ - for each h_j, look at the sublevel set
+ { v : f_y(v) < g_y(h_j) + eps' }, an interval;
+ - inside each interval, choose nested rectangles of
+ prescribed areas h_1 < h_2 < ... < h_l;
+ - smooth the rectangle corners into nested smooth
+ disks of the same area;
+ - Schlenk's "nested disk lemma" (a standard trick)
+ gives a Hamiltonian isotopy carrying the standard
+ concentric disks of areas h_j to the chosen disks;
+ - now any point of radial area v sits below the "next
+ grid point up," and evaluating f there is at most
+ g(v) + 2 eps'.
+
+ Lifting to C^m: for the j-th factor, apply the planar
+ construction to f_y(v) = F(y, L(y) v), and lift the
+ Hamiltonian by l(y) * B_t(y, z_j / sqrt(l(y))) where
+ l(y) is a smooth approximation to the slice length L(y)
+ from below. On the outer strip L(y) < p_j, the maps can
+ be taken to be the identity (there is nothing to sort
+ there), which keeps the construction smooth over Delta.
+
+ Because the planar Hamiltonian vanishes near the boundary
+ of the disk, the lift vanishes near the outer face
+ b_nu . p = c_nu. This is the "outer face preservation"
+ property used in the Moser step later.
+
+ Summing the errors over j = 1, ..., m gives
+
+ (K - P) o q <= R(K - P) + sum_j eps_j
+ <= K - d + sum_j eps_j
+ < K
+
+ provided sum eps_j < d.
+
+ Finally approximate K by a smooth function; the error is
+ smaller than d/4, so the strict gap survives.
+
+QED (comparison lemma).
+
+WHAT WE GET.
+ From a mean inequality, a strict pointwise inequality. From
+ a convex function P, we get a smooth toric K and a Hamiltonian
+ q that turn the mean surplus into a pointwise surplus.
+
+ The next section uses that surplus to cut disjoint layers.
+
+
+6. STEP B: PACKING OVER A HANDLE
+----------------------------------------------------------------
+
+Now we actually place the balls. The setup is a base surface
+C^o of genus >= 1 (with finitely many points removed, at least
+one) times the toric fiber V from Section 5. A positive-genus
+surface has a "handle" -- an essential circle that cannot be
+shrunk to a point. That handle is what allows us to make
+disjoint layers.
+
+THE MODEL.
+ We have a family of two-forms on Sigma x V, one for each
+ compact bordered surface Sigma inside C^o. Each has the
+ form
+
+ Omega = omega_V + d Gamma,
+
+ with Gamma a toric horizontal one-form (annihilates vectors
+ tangent to V, coefficients depending smoothly on both base
+ point and moment p). The base curvature
+
+ d_C Gamma = kappa(p) dp_1 ^ dp_2
+
+ is positive, and its integral over Sigma,
+
+ A(p) = int over Sigma of kappa(p),
+
+ is assumed to converge uniformly to a continuous positive
+ limit H(p) as Sigma grows.
+
+ This "surface form" is exactly what the geometric models of
+ Section 7 provide.
+
+THE LEMMA (surface packing).
+ Let r_1, ..., r_k > 0 be positive numbers, each simplex
+ { p >= 0 : sum_j p_j <= r_i } strictly inside Delta and
+ strictly away from every outer face. Define the leftover
+ function
+
+ P(p) = H(p) - sum_i (r_i - sum_j p_j)_+ .
+
+ Suppose:
+ (i) P is concave;
+ (ii) P has positive average over Delta;
+ (iii) P is positive on Delta minus tau Delta.
+
+ Then for any capacities r_i' < r_i, some model (i.e. some
+ sufficiently large Sigma) contains a compact packing of the
+ closed balls of capacities r_i' in the interior of
+ Sigma x V, with neighborhood embeddings.
+
+ That is: the three hypotheses turn into an actual packing
+ into the model.
+
+PROOF, IN FIVE STEPS.
+
+ STEP 1: shrink to a finite model.
+ Hamiltonian comparison (Section 5) provides K and q
+ with (K-P)(qu) < K(u). The uniform gap is
+
+ g = min_v [ K(q^{-1}v) - K(v) + H(p(v)) - C_r(p(v)) ]
+
+ where C_r(p) is the cap sum. g > 0 by the lemma's
+ hypotheses.
+
+ Choose 0 < eps < g/8 (this is very small). Take Sigma
+ so large that
+ |A(p) - H(p)| < eps
+ for all p in Delta (this is possible by uniform
+ convergence of A to H). From now on, "A" means the
+ finite base area, "H" the limiting area.
+
+ We will use the comparison based on the LIMITING area H,
+ not the finite A. This is why the lemma is stated in
+ terms of H.
+
+ STEP 2: place the base area on an annulus.
+ Inside the handle choose an annular region A^+ with a
+ positive coordinate s (going around the annulus) and an
+ angular coordinate t in R/Z. A closed one-form
+ beta on the base is supported in a transverse annulus
+ B, and on A^+ looks like
+
+ beta|A^+ = f(t) dt, f >= 0, integral f = 1.
+
+ So beta has period exactly 1 around the essential
+ circle. This is the "handle form"; its integral around
+ the handle is 1.
+
+ Now modify the base area. Pick a smooth smaller area
+ kappa_sm(p) that (a) agrees with kappa(p) near the base
+ boundary, (b) is independent of p on A^+, and (c) has
+ small total integral:
+
+ sup_p int_Sigma kappa_sm(p) < eps.
+
+ Let
+ H_*(p) = int_Sigma ( kappa(p) - kappa_sm(p) ),
+ kappa_1(p) = kappa_sm(p) + H_*(p) chi_A,
+
+ where chi_A is a bump on the annulus A with integral 1.
+
+ Then kappa_1 > 0, |H_* - H| < 2 eps, kappa_1 - kappa has
+ zero total integral. By a standard "relative area
+ primitive" lemma, we can normalize the base one-form
+ Gamma so that on the annulus
+
+ d_C Gamma_1 = kappa_1, Gamma_1|A = gamma + s H_* beta,
+
+ where d gamma = kappa_sm on the annulus. The total form
+ is still symplectic.
+
+ STEP 3: build the "layer path."
+ Choose smooth positive toric functions h_i on Delta with
+
+ h_i(p) > (r_i - sum_j p_j)_+,
+ 0 < sum_i h_i - C_r < eps.
+
+ (This makes h_i slightly larger than the "cap" function
+ for the i-th ball.)
+
+ Define
+ J(v) = K(v) - H_*(p(v)),
+ R_*(v) = K(q^{-1}v) - J(v) - sum_i h_i(p(v)).
+
+ Then R_* >= g - 3 eps > 0 by the gap.
+
+ The gap says: on each fiber v, the s-coordinate (going
+ around the handle) can interpolate a certain coefficient
+ from J(v) at the lower end of the handle to K(q^{-1}v)
+ at the upper end, in steps of h_i(p(v)) with positive
+ leftover R_*(v).
+
+ Concretely, define a family of forms on A x V by
+
+ Omega_lambda = d gamma + D_lambda^*(omega_V + d(G_lambda beta))
+
+ where D_lambda is a Hamiltonian isotopy of the fiber
+ (moving from identity to q as the s-coordinate runs
+ from 0 to 1), and G_lambda is a scalar function on
+ A x V determined by the interpolation. The suspension
+ formula (standard in symplectic topology) shows that
+ Omega_lambda is symplectic whenever the s-derivative
+ of the scalar part is nonnegative.
+
+ At the ends, the form matches the model from Step 2,
+ so Omega_lambda extends globally.
+
+ Refine G from "just interpolating" to "interpolating
+ with marked layers": choose disjoint sub-intervals
+ I_1, ..., I_k of the s-axis, and a residual tail I_*.
+ Along I_i, increment G by h_i(p(v)) alone. Then the
+ form on layer i is
+
+ omega_V + d(G_f beta) with G_f = a_i(p) + rho_i(s) h_i(p).
+
+ The "a_i(p)" part contributes a toric Hamiltonian to the
+ fiber, and the "rho_i(s) h_i(p)" part contributes the
+ linear growth in s.
+
+ STEP 4: restore the original form via Moser.
+ The modified form is symplectic and has the same
+ boundary conditions on the base boundary. But we want
+ it back on the original model. Use a Moser isotopy:
+
+ - the family of forms has the same cohomology class
+ at each stage,
+ - the family preserves every outer face b_nu . p
+ (because the construction uses toric functions and
+ outer-face-commuting Hamiltonians),
+ - the base boundary collar is fixed.
+
+ The "Moser with outer faces" lemma guarantees an
+ isotopy that undoes the deformation while keeping the
+ base boundary collar fixed and (crucially) never
+ pushing any trajectory across an outer face.
+
+ So the final picture: a diffeomorphism psi from the
+ "layer-modified" model back to the original model.
+
+ STEP 5: embed one closed ball in each layer.
+ In the i-th layer, the form is exactly
+
+ omega_V(v') + dx ^ dT
+
+ after a coordinate change, where:
+ (s, T) are base coordinates,
+ x = L(s,T) + rho_i(s) h_i(p) (a base function),
+ v' = fiber coordinates with the p-parameters.
+
+ The region of this product form that is available in
+ the i-th layer is
+ 0 < T < 1, 0 < x < h_i(p(v')).
+
+ Now we want to fit a closed ball of capacity r_i' < r_i.
+ A planar trick (a corollary of the nested-disk lemma):
+ for any A > r_i' and delta > 0, a neighborhood of the
+ closed disk of area A in the (x, T) plane embeds
+ symplectically so that
+
+ point of radial area w sits at
+ x < w + delta, 0 < T < 1.
+
+ Choose delta < r_i - r_i'. Then for every point of the
+ ball,
+ w + sum_j p_j(v') <= r_i' (by the ball's
+ definition),
+ x < w + delta < r_i - sum_j p_j(v') (subtract),
+ hence x < h_i(p(v')) (by h_i > cap).
+
+ So the ball fits strictly inside the layer with room to
+ spare. The layer is compactly contained in the strip,
+ and the strip is away from the outer faces. So the
+ closed ball is inside the layer with compact closure,
+ and the embedding extends to a neighborhood.
+
+ Different layers have disjoint base intervals I_i, so
+ their fiber images are automatically disjoint.
+
+ Compose with psi^{-1} from Step 4. Done: we have a
+ packing of the k closed balls inside the original model.
+
+WHAT WE GET.
+ From the mean inequality of Section 5 and the handle of a
+ positive-genus surface, we get a concrete packing in an
+ explicit surface-plus-toric model. The packing stays away
+ from the outer faces of Delta, so it survives any further
+ deformation that respects those faces.
+
+ Section 7 builds such models inside the symplectic ball.
+
+
+7. STEP C: GEOMETRIC MODELS AND TRANSFER
+----------------------------------------------------------------
+
+Now we have to build surface-plus-fiber models inside the
+target ball, and to move packings between them. The four
+main tools live in geometry.tex.
+
+7.1 RELATIVE KÄHLER MOSER
+
+ Classical Moser's method: given a smooth path of symplectic
+ forms omega_t = omega_0 + d alpha_t with all omega_t
+ symplectic and nondegenerate, integrate the vector fields
+ X_t defined by
+ iota_{X_t} omega_t = -alpha_t
+ into an isotopy psi_t with psi_t^* omega_t = omega_0.
+
+ On a compact complex manifold, this shows: two Kähler forms
+ in the same cohomology class are related by an isotopy
+ preserving the Kahler property.
+
+ The paper's RELATIVE version: if you also want to
+ preserve a list of disjoint smooth complex submanifolds
+ S_1, ..., S_e as SETS, you can arrange that psi_t(S_j) = S_j.
+ The trick: along each S_j, prescribe the first jet of a
+ corrector function f_t so that the Moser field is tangent to
+ S_j.
+
+ This is the tool that lets us "leak" past forbidden
+ divisors when we come back to the target ball. When we say
+ "remove H_infty from PP^n," we mean "after applying a
+ Moser isotopy that preserves both H_infty and the blowup
+ divisor E."
+
+7.2 NORMAL-CONE TRANSFER
+
+ Setup. Let T be a smooth projective variety, Z a smooth
+ closed submanifold, L an ample rational polarization. Let
+ T' = Bl_Z T, Y = PP_Z( 1 + N_{Z/T} ),
+ where N_{Z/T} is the normal bundle of Z inside T, and
+ PP_Z(1 + N_{Z/T}) is the projective bundle obtained by
+ adding a trivial line to N_{Z/T}.
+
+ Choose rational d > 0. Suppose two classes are ample:
+
+ on T': pi^* L - d E_Z,
+ on Y: L|_Z + d U.
+
+ (Here E_Z is the exceptional divisor and U is the dual
+ tautological class.)
+
+ THE LEMMA.
+ Let F subset T be a finite union of disjoint smooth
+ complex submanifolds (a set we want to keep avoiding).
+ Let omega_T be any Kähler form in class L, and omega_Y
+ any Kähler form in class L|_Z + d U invariant under
+ scalar rotation of the normal summand of Y.
+
+ Then every compact subset of
+ Y minus ( D_infty union pi_Y^{-1}(F cap Z) )
+ has a neighborhood admitting a symplectic embedding
+ into
+ ( T minus F, omega_T ),
+ with its given form omega_Y.
+
+ Here D_infty = PP_Z(N_{Z/T}) is the "infinity
+ section" of Y.
+
+ Idea of the proof.
+ Use the CLASSICAL construction of deformation to the
+ normal cone:
+
+ X = Bl_{Z x {0}} ( T x A^1 ) -> A^1.
+
+ Its fiber at t != 0 is T; its fiber at t = 0 is the
+ union of T' and Y, glued along D_infty.
+
+ Put a rational polarization L - d [Y] on the total
+ space X. It restricts to pi^* L - d E_Z on T' and
+ to L|_Z + d U on Y. Ampleness on each component
+ plus finiteness of the map
+ T' u Y (disjoint) -> central fiber
+ implies ampleness on the central fiber, which is an
+ open condition; so the total polarization is ample
+ near zero.
+
+ For any compact K in Y away from D_infty and
+ pi_Y^{-1}(F), the map Y -> T along the deformation
+ (i.e. flowing from Y into a nearby smooth fiber T_t)
+ is defined on a neighborhood of K. That flow is
+ our symplectic embedding.
+
+ In short: deformation to the normal cone is a "video"
+ that starts at Y (at t = 0) and ends at T (at t -> 0 but
+ nonzero). The construction carries symplectic structures
+ along the video.
+
+ Intuition for why this is useful. A projective manifold
+ has a "zoom into Z" limit where it becomes the projective
+ bundle Y. If Y is a nicer target (e.g. a toric model over
+ a curve), we can do the packing there and zoom back. This
+ is how the paper transfers packings from the "surface plus
+ toric fiber" model into the actual ball.
+
+7.3 EXPLICIT NORMAL-BUNDLE FORM
+
+ Suppose Z is a curve (complex dimension 1) and
+ N_{Z/T} = A^b + A^b + ... + A^b (l summands),
+ where A is an ample line bundle on Z with Chern curvature
+ sigma > 0, and b > 0 is an integer. If bd < 1, the
+ class sigma + d U on Y = PP_Z(1 + N) has an explicit
+ Kahler representative of the shape
+
+ omega_Y = sigma + d( sum_{j=1}^l p_j alpha_j ),
+ p_j >= 0, sum_j p_j < d,
+
+ where alpha_j is the angular connection on the j-th normal
+ line (period one, curvature -b sigma), and p_j are the
+ standard fiber moments.
+
+ The (p_1, ..., p_l) coordinates are nonnegative and their
+ sum is bounded by d. So the fiber over each point of Z is
+ a polytope of ALLOWED moment values, which is exactly the
+ "Delta" of Section 5's toric domain V. This is the
+ precise sense in which Y is a "surface plus toric fiber"
+ model.
+
+ For Z = C a curve and Sigma subset C a bordered subsurface,
+ the model restricted to Sigma x V has a toric horizontal
+ primitive Gamma with base curvature
+
+ kappa(p) = ( 1 - b sum_j p_j ) sigma.
+
+ The integral of kappa over Sigma tends to (deg A) times
+ ( 1 - b sum_j p_j ) as Sigma grows towards C. That is
+ exactly the limiting base area H(p) used by the surface
+ packing lemma (Section 6).
+
+ So the normal-bundle model Y is precisely a surface-plus-
+ fiber model, with H(p) = (deg A) (1 - b sum p_j). This is
+ the reason the constructions work.
+
+7.4 BLOWUP COMPLEMENT IS A SHELL
+
+ Let n >= 2, o a point of PP^n, H_infty a hyperplane not
+ containing o. Let T = Bl_o PP^n, E the exceptional
+ divisor, H the pullback of the hyperplane class of PP^n.
+
+ For 0 < a < 1, the complement T minus (E union H_infty)
+ with its standard Kähler form in class H - aE is
+ symplectomorphic to the shell
+
+ { z in C^n : a < pi sum_j |z_j|^2 < 1 }.
+
+ Proof sketch. Use the "symplectic cut" construction
+ (Lerman 1995): act on PP^n x C by a circle, reduce at a
+ suitable level, and you get exactly the blowup T with its
+ standard form. Removing E and H_infty leaves the shell.
+ A direct model-check then matches the affine Fubini-Study
+ coordinates to standard complex coordinates in C^n.
+
+ This is the punchline: "blowup a point, remove two
+ divisors, get the shell." The shell is precisely what
+ Cases 2 and 3 need for the reserved-ball reduction.
+
+7.5 PUTTING IT TOGETHER
+
+ A model for the target often looks like this:
+
+ PP^n -> (blow up at o) T = Bl_o PP^n -> D subset T
+ (a curve or
+ a hyperplane)
+ |
+ v
+ Y = PP_D(1 + N_{D/T})
+
+ with the class L|_D + d U on Y. That class is ample by
+ choosing d small.
+
+ The surface-packing lemma (Section 6) works inside Y when
+ we can identify Y's base D as a punctured curve of positive
+ genus. So:
+
+ - pack into Y,
+ - transfer back to T via normal-cone transfer,
+ - identify T minus (E u H_infty) with a shell in C^n
+ via the blowup-complement lemma,
+ - add the reserved central ball if applicable.
+
+ That is the whole pipeline. Sections 8 flesh it out for
+ the three cases.
+
+
+8. THE THREE CASES IN DETAIL
+----------------------------------------------------------------
+
+Normalize the target capacity to 1. In every case, we produce
+an ample model with base a punctured positive-genus curve, whose
+limiting base area H(p) has all the required properties.
+
+8.1 CASE 1: ALL CAPACITIES BELOW 1/2
+
+ Take PP^n. Choose a hyperplane H_infty (a generic one). Pick
+ n-1 quadrics Q_1, ..., Q_{n-1} in general position. Their
+ intersection
+
+ C = Q_1 cap Q_2 cap ... cap Q_{n-1} subset PP^n
+
+ is a smooth connected curve of degree 2^{n-1} (Bertini's
+ theorem ensures this for a generic choice of quadrics). Its
+ genus is
+
+ g(C) = 1 + (n-3) * 2^{n-2},
+
+ which is 1 for n=3, 5 for n=4, 17 for n=5, and so on. In any
+ case, POSITIVE for every n >= 3.
+
+ Its NORMAL bundle in PP^n is ( O_C(2) )^{oplus (n-1)}: one
+ degree-2 line bundle per quadric. Each quadric gives a
+ section of O(1) that vanishes precisely on C, so the normal
+ direction of C inside Q_1 cap ... cap Q_j has class the
+ corresponding O(2).
+
+ Deform to normal cone. T = PP^n, Z = C, L = O(1). The
+ resulting model is
+
+ Y = PP_C( 1 + O_C(2)^{oplus (n-1)} ),
+
+ with class L|_C + d U. Ampleness of the two components
+ (Bl_C PP^n with pi^*L - d E_C, and Y with L|_C + d U) holds
+ for 0 < d < 1/2, which is standard (adjunction in
+ PP^n).
+
+ The base area (from the explicit normal-bundle form, Section
+ 7.3) is
+
+ H_0(p) = 2^m ( 1 - 2 sum_{j=1}^m p_j ), m = n-1.
+
+ (The 2^m is deg O_C(1) = 2^{n-1} = 2^m, the degree of C as
+ a curve in PP^n. The factor (1 - 2 sum p_j) is 1 - b sum p_j
+ with b = 2, one per summand of the normal bundle O_C(2).
+ Also m = n-1 is the number of normal summands.)
+
+ The truncation is at sum p_j <= h for small h. Its integral
+ is
+
+ J_m(h) = int_{sum p_j <= h} 2^m ( 1 - 2 t ) dp
+ = (2^m h^m / m!) ( 1 - 2mh / (m+1) )
+
+ which tends to 1 / n! as h -> 1/2.
+
+ And 1 / n! is EXACTLY the volume of the unit ball B^{2n}(1).
+ So as h approaches 1/2, the model captures ALL of the volume.
+
+ Fixing parameters. Given requested capacities R_1, ..., R_k
+ all < 1/2 with sum R_i^n < 1, choose rationals
+
+ r_* < h < d < 1/2,
+
+ where r_* = max r_i with each r_i chosen in
+ (R_i, 1/2), and J_m(h) > sum r_i^n / n!.
+
+ These exist because J_m(h) -> 1/n! as h -> 1/2, and the
+ volume inequality is strict.
+
+ Now define P(p) = H_0(p) - sum_i (r_i - t)_+, t = sum p_j.
+ Check:
+ concavity: H_0 is affine in t, each cap (r_i - t)_+
+ is convex, so P is concave.
+ mean positive: int P dp = J_m(h) - sum r_i^n/n! > 0.
+ boundary: taking tau with r_*/h < tau < 1, we
+ have that for p outside tau Delta,
+ all caps vanish, and H_0(p) >= positive.
+ So all hypotheses of the surface packing lemma hold.
+
+ Apply surface packing lemma (Section 6) with parameters
+ r_i' < r_i. Get a compact packing of the closed balls of
+ capacities r_i' inside one model Y.
+
+ The packing lives away from D_infty (the infinity section
+ of Y) and away from pi_Y^{-1}(C cap H_infty). Therefore
+ it transfers back into PP^n minus H_infty via normal-cone
+ transfer.
+
+ PP^n minus H_infty is the open ball B^{2n}(1) via
+ affine Fubini-Study coordinates (the map
+ z -> z / sqrt(pi (1 + sum |z|^2))
+ is a symplectomorphism). So the packing lands in the
+ desired target.
+
+ Neighborhood statement: all constructions are compact and
+ defined on neighborhoods; shrink at each composition. Since
+ R_i < r_i' < r_i, we have the required slack.
+
+8.2 CASE 2: ONE LARGE CAPACITY, DIMENSION n >= 4
+
+ Let A >= 1/2 be the large capacity. Choose rationals
+
+ 1/2 < a < 1, A < a, s = 1 - a.
+
+ The reserved central ball has capacity A < a. The remaining
+ balls (R_1, ..., R_k) must fit into the shell
+ { a < pi sum |z_j|^2 < 1 }.
+
+ For each i, choose r_i > R_i so that r_i < s and
+ sum r_i^n < 1 - a^n.
+
+ Construct a "two-stage" model.
+
+ Stage 1: PP^n -> hyperplane D = PP^{n-1} -> normal cone.
+ Let T = Bl_o PP^n, L = H - aE. Choose a hyperplane D
+ not containing o. Deform to normal cone of D. Component
+ class on Bl_D T is L - d_1 D, ample for d_1 < s (this
+ is standard for blowups at a point of projective space).
+ Component class on Y_1 = PP_D(1 + O_D(1)) is L|_D +
+ d_1 U_1, ample for d_1 < 1 - a = s. Pick rationals
+ r_* < h_1 < d_1 < s.
+
+ The base area function is
+ H_0^(1)(u) = 1 - u, u = first moment on D,
+ i.e. deg(O_D(1)) = 1.
+
+ Stage 2. Inside D = PP^{n-1}, run Case 1 with the
+ appropriate dimension (n-1 instead of n, so n-1 >= 3
+ requires n >= 4). The curve
+
+ C = intersection of (n-2) generic quadrics in D
+
+ is smooth of genus g(C) = 1 + (n-4) * 2^{n-3}, which
+ is 1 for n=4, 5 for n=5, 17 for n=6, and so on. Positive
+ for every n >= 4.
+
+ The stage-2 model is Y_2 = PP_C(1 + O_C(2)^{oplus
+ (n-2)}), with base area
+ H_0^(2)(v) = 2^{n-2} ( 1 - 2 sum_{j=2}^{n-1} v_j ),
+ v = (v_2, ..., v_{n-1}) the second-stage moments.
+
+ The truncation parameters are r_* < h_2 < d_2 < 1/2.
+
+ Combining the two stages. The stage-2 model pulls back
+ into the stage-1 fiber (Stage 1 gives a ball for the first
+ normal direction over D, then Stage 2 gives another toric
+ ball inside D). After the "moment change" coordinate
+ (z_1, z_2, ..., z_{n-1}) with u = pi |z_1|^2,
+ v_j = pi |z_j|^2 (j >= 2), define
+
+ zeta_1 = z_1, zeta_j = sqrt(1 - u) z_j (j >= 2),
+
+ with new moments
+
+ p_1 = u, p_j = (1 - u) v_j (j >= 2).
+
+ This is a smooth diffeomorphism on u < 1 including all axes.
+ Set w = p_2 + ... + p_{n-1}. The combined polytope is
+
+ Delta(h_1, h_2) = { p >= 0 : p_1 <= h_1,
+ w + h_2 p_1 <= h_2 }.
+
+ Combined base area function (verified in Section 5 of
+ applications.tex):
+
+ H_0(p) = 2^{n-2} ( 1 - p_1 - 2w ).
+
+ Its coefficient (1 - p_1 - 2 w) stays positive on Delta
+ because Delta has p_1 <= h_1 < 1 and w + h_2 p_1 <= h_2,
+ so w <= h_2 (1 - p_1) and 1 - p_1 - 2w
+ >= (1 - p_1)(1 - 2 h_2) >= (1 - h_1)(1 - 2 h_2) > 0
+ when h_1 < 1 and h_2 < 1/2.
+
+ Volume. The integral of H_0 over Delta(h_1, h_2) is
+
+ B_q(h_2) * ( 1 - (1 - h_1)^n ) / n,
+
+ where
+ B_q(h_2) = 2^q ( h_2^q / q! - 2q h_2^{q+1} / (q+1)! )
+ -> 1 / (q+1)! as h_2 -> 1/2,
+
+ q = n - 2. So (B_q vol) -> ( 1 - a^n ) / n! as
+ h_1 -> s and h_2 -> 1/2.
+
+ And (1 - a^n) / n! is EXACTLY the volume of the shell
+ { a < pi sum |z_j|^2 < 1 }.
+
+ Same concavity / mean / outer-region checks as Case 1. The
+ cap functions are (r_i - p_1 - w)_+ and the "+" is still
+ concave. Choose truncations so that int P dp > 0, and
+ r_*/h_1, r_*/h_2 are both < tau < 1. Then P is positive
+ on Delta minus tau Delta.
+
+ Apply surface packing lemma to get a compact packing inside
+ Y_2. Then transfer:
+
+ Y_2 -> (embed via phi into) D minus (C cap H_infty)
+ -> (transfer to) Y_1 minus (its infinity + bad set)
+ -> (transfer to) T minus (E union H_infty)
+ -> (identify) shell in C^n.
+
+ Add the reserved central ball of capacity A < a at the
+ center. All disjoint, all with compact closure, all with
+ neighborhood embeddings.
+
+8.3 CASE 3: ONE LARGE CAPACITY, DIMENSION n = 3
+
+ Same setup as Case 2, but n = 3 so the plane inside PP^3
+ is PP^2, and a conic in PP^2 has genus 0. No handle.
+
+ Replace conic by a plane CUBIC curve C through two
+ distinguishable points q_1, q_2. Such C has genus 1.
+ Moreover, C minus { q_1, q_2 } still has (non-compact)
+ genus 1. So the surface packing lemma still works, with
+ C^o = C minus { two points and possibly extra points }.
+
+ Extra wrinkle: we need to cut the moment at p_1 = c
+ for c < s, to allow Stage 1 and Stage 2 to fit together.
+ This is a KÄHLER CUT.
+
+ Setup for the cut.
+ V_hat = Bl_{(q_1,0), (q_2,0)} PP_{PP^2}(1 + O_{PP^2}(3)).
+ This is a projective bundle over PP^2 with a line
+ bundle O(3), blown up at two points.
+
+ The standard circle rotating the O(3) summand lifts to
+ V_hat. Its Hamiltonian is p_1 (the first moment).
+ Normalize p_1 = 0 on the "bottom" (the strict transform
+ of the zero section).
+
+ The fixed loci of the circle are at levels:
+ 0, s, d_1,
+
+ where s = 1 - a (mass of the exceptional curves) and
+ d_1 the ordinary fiber mass. Both are positive for
+ 0 < c < s.
+
+ The cut.
+ Take the Kähler reduction of V_hat x C by the
+ diagonal circle at level
+ p_1 + pi |v|^2 = c.
+
+ Because c < s, every point of the level lies over the
+ attracting basin of the bottom section. The complex
+ quotient is a projective bundle
+
+ Y = PP_D( 1 + N_{D/V_hat} ), D = Bl_{q_1,q_2} PP^2,
+
+ with class L|_D + c U_c. This Y is a projective bundle
+ over D (a plane surface blown up at two points) with
+ normal bundle
+
+ N_{D/T} = O_D(3M - e_1 - e_2),
+
+ in classes M = pullback of plane hyperplane class,
+ e_i = exceptional divisors over q_1, q_2.
+
+ Ample classes: L = 3b M - s (e_1 + e_2) with
+ b = (2 - a)/3 = (1 + s)/3, s = 1 - a, and L is the
+ class on D.
+
+ Now Stage 1 is a quadric-surface normal model (different
+ from Case 2's hyperplane):
+ D = Bl_{q_1,q_2} PP^2,
+ Y_1 = PP_D(1 + O_D(3M - e_1 - e_2)).
+ Stage 2 is a cubic curve model living in D:
+ C = strict transform in D of a smooth plane cubic
+ through q_1, q_2.
+ Y_2 = PP_C(1 + O_C(3)) is a toric model over C.
+ deg_C O(3) = 9.
+
+ Base area function (Section 4 of sixdim.tex).
+ The full projective system on the cubic has the
+ curve-vanishing weight computed as follows. Two
+ marked points, each contributing one logarithmic
+ mass (s - p_1 - p_2)_+ to the base area. And
+ deg O_C(3) = 9, so the base area is
+
+ H_0(p) = 9 ( b - p_1 - p_2 ) - 2 ( s - p_1 - p_2 )_+.
+
+ Here b = (1 + s) / 3 as above.
+
+ The 9 b is the constant coefficient at zero moment:
+ it comes from deg O_C(3) = 9 times the parameter b.
+ The -9 p_1 - 9 p_2 is the base area loss per unit
+ moment. The -2(s - t)_+ is the loss due to the two
+ marked-point atoms, each missing mass (s - t)_+.
+
+ Volume.
+ Integrate over Delta_infty = { 0 <= p_1 <= s,
+ 0 <= p_1 + p_2 <= b }:
+
+ int H_0(p) dp = (1 - a^3) / 6,
+
+ which is EXACTLY the volume of the shell
+ { a < pi sum |z_j|^2 < 1 } in complex dimension 3.
+
+ (The computation is direct integration in one variable.)
+
+ Concavity of H_0.
+ As a function of t = p_1 + p_2, H_0 is piecewise affine
+ with slopes -7 for t < s and -9 for t > s. Concave.
+
+ Apply surface packing lemma with truncated polytope
+
+ Delta = { p >= 0 : p_1 <= h_1, p_1 + p_2 <= h_2 },
+
+ for h_1 near s and h_2 near b. The cap supports are
+ inside Delta; there is enough volume for all balls.
+
+ Transfer chain:
+
+ cubic model Y_2 ->
+ moment-preserving parallel transport into V_hat
+ (p_1 <= c preserved exactly) ->
+ lower Kähler cut at p_1 = c ->
+ Y_1 = PP_D(1 + O_D(3M - e_1 - e_2)) ->
+ normal-cone transfer into T = Bl_o PP^3 minus
+ (E union H_infty) ->
+ identified with shell in C^3 ->
+ add reserved central ball of capacity A < a.
+
+ All disjoint, all compact, all with neighborhood embeddings.
+
+8.4 WHY THESE THREE COVER EVERYTHING
+
+ Given R_1, ..., R_k with sum R_i^n < 1 and all pairs
+ R_i + R_j < 1:
+
+ - if k = 1, put it by inclusion.
+ - if all R_i < 1/2, use Case 1.
+ - otherwise exactly one is >= 1/2 (uniqueness from
+ the two-ball inequality). Call it A.
+ - if n >= 4, use Case 2.
+ - if n = 3, use Case 3.
+
+ In every case, the assumed hypotheses give exactly the
+ properties needed (concavity, positive mean, boundary gap).
+ So the sufficiency direction of the theorem is proven.
+
+
+9. WHY DIMENSION FOUR IS DIFFERENT
+----------------------------------------------------------------
+
+The theorem is for n >= 3, i.e. 6 real dimensions and up. What
+about n = 1 (2 real dimensions) and n = 2 (4 real dimensions)?
+
+n = 1 (ordinary plane, real dimension 2):
+ Symplectic = area preserving. Packing is purely a matter
+ of area; the criterion is just sum R_i < R. No pairwise
+ condition needed. Nothing interesting.
+
+n = 2 (real dimension 4):
+ This is the richest case. Gromov proved the two-ball
+ obstruction in 1985. McDuff and Polterovich (1994) then
+ showed that volume + pairwise is NOT sufficient: there
+ are infinitely many additional obstructions, coming from
+ exceptional classes in blowups of CP^2 and their
+ J-holomorphic curves. These are families of "rigid rulers"
+ that further restrict packing configurations.
+
+ Biran (1997, 1999) established "packing stability" in this
+ dimension: on a closed symplectic four-manifold with rational
+ symplectic class, any sufficiently large number of EQUAL
+ balls can fill an arbitrarily large fraction of the volume.
+ Buse-Hind (2011, 2013) extended equal-ball stability to
+ higher-dimensional balls and to all closed rational
+ symplectic manifolds. Buse-Hind-Opshtein (2016) proved
+ strong stability for unequal balls, provided all capacities
+ are sufficiently small.
+
+ But the sharp criterion (exactly which configurations fit)
+ remains an open, and much harder, question in dimension 4.
+
+n >= 3 (6 real dimensions and up):
+ The extra dimensions give "floppiness." The paper proves
+ that volume + two-ball IS sufficient. All other
+ obstructions disappear. So the answer is exactly:
+
+ pack iff (sum R_i^n) < R^n AND (all R_i+R_j < R).
+
+This is a strange but common phenomenon in high-dimensional
+geometry: things that are rigid in low dimensions become
+flexible once there is enough room.
+
+
+10. WHAT THE PROOF DOES AND DOES NOT GIVE
+----------------------------------------------------------------
+
+The proof is a pure existence proof. Specifically:
+
+ It tells you a packing EXISTS.
+ It does NOT tell you an algorithm for constructing one.
+ It does NOT tell you the shapes of the packed balls.
+ It does NOT give numerical effective bounds on how small
+ the neighborhoods around the closed balls can be.
+
+That last point is why every "embedding" here means "symplectic
+embedding defined on an open neighborhood of the closed source
+ball." This is the standard convention, but it's worth
+being explicit about.
+
+What the proof does open up:
+
+ - The surface-plus-toric-fiber machine (Sections 5-6) is
+ quite general. Similar constructions (symplectic folding,
+ Moser-type normalizations, positive-genus bases) appear
+ across symplectic topology.
+ - The interplay between packing and the algebraic structure
+ of the Hamiltonian diffeomorphism group (Edtmair 2025)
+ was a motivation. The paper's argument doesn't use those
+ results, but the connection is real.
+ - The normal-cone transfer with a lower Kähler cut
+ (Section 7.2 / 8.3) is a reusable device.
+
+What the proof does NOT settle:
+
+ - Dimension four. As discussed, the criterion here is
+ different (richer set of obstructions) and is still an
+ active research area.
+ - Stabilized packing: packings after crossing with a fixed
+ symplectic surface. Siegel and Yao's original paper
+ studied these too; they are not resolved by the present
+ theorem.
+ - Complex dimension 1 and 2 - trivially resolved and
+ richer-respectively, as noted above.
+
+
+11. GLOSSARY
+----------------------------------------------------------------
+
+Ball of capacity R:
+ B^{2n}(R) = { z in C^n : pi sum |z_j|^2 <= R }.
+ Radius sqrt(R/pi). Volume R^n / n!.
+
+Capacity, symplectic:
+ The natural "area" unit. A capacity-one ball has a complex
+ line through the origin of symplectic area exactly one.
+
+Symplectic form omega_0:
+ The real 2-form sum_j dx_j ^ dy_j on C^n = R^{2n}. It
+ measures signed area on each complex coordinate plane and
+ sums those areas.
+
+Symplectic embedding:
+ A smooth injection f : U -> U' with f^* omega_0 = omega_0.
+
+Volume-preserving embedding:
+ A smooth injection with f^* vol = vol. Much weaker than
+ symplectic in dimension >= 2n >= 4. The gap between the
+ two is what symplectic geometry is about.
+
+Gromov's two-ball obstruction:
+ In any n >= 2, symplectic balls of capacities R_1, R_2
+ cannot both fit disjointly into capacity R unless
+ R_1 + R_2 < R.
+
+J-holomorphic curve:
+ A map from a Riemann surface (usually S^2) into a
+ symplectic manifold that is holomorphic with respect to a
+ compatible J. These are the "rigid rulers" that detect
+ non-volume obstructions.
+
+Conjecture A (Siegel-Yao 2025):
+ For n >= 3, volume + two-ball is sufficient. Now a
+ theorem.
+
+Toric domain:
+ V = { z in C^m : p(z) in Delta } for a "downward" polytope
+ Delta in the nonnegative orthant. Standard form:
+ omega_V = sum dp_j ^ d theta_j.
+
+Outer face:
+ A face of Delta of the form b_nu . p = c_nu with c_nu > 0.
+ The coordinate faces p_j = 0 are not outer.
+
+Hamiltonian comparison (Section 5):
+ The lemma that upgrades a positive-mean inequality into a
+ strict pointwise inequality via increasing rearrangements
+ and a contractive fixed-point argument. Output: K (toric)
+ and q (Hamiltonian) with (K-P) o q < K.
+
+Increasing rearrangement:
+ Given a convex function f on [0,1], its increasing
+ rearrangement g is the (unique up to values) function with
+ g nondecreasing and the level sets of g having the same
+ length as those of f. Formula:
+ g(h) = min_{0 <= a <= 1-h} max{ f(a), f(a+h) }.
+
+Surface packing lemma (Section 6):
+ Packs disjoint closed balls in a surface-plus-toric model,
+ given a concave P with positive mean and a boundary
+ positivity property, and given a positive-genus punctured
+ base with a handle-form beta of period 1.
+
+Handle form beta:
+ A closed one-form on the surface with int(beta) = 1 around
+ an essential circle in the handle. This is the mechanism
+ that creates disjoint "layers" along one coordinate.
+
+Deformation to the normal cone:
+ A one-parameter family over A^1 whose fiber at t != 0 is
+ a given T and whose fiber at t = 0 splits into the blowup
+ Bl_Z T and the projectivized normal bundle
+ PP_Z(1 + N_{Z/T}).
+ Used to move packings between these pieces.
+
+Normal-cone transfer (Section 7.2):
+ The lemma that transports a compact packing from
+ PP_Z(1 + N_{Z/T}) into T itself, via the deformation to
+ the normal cone, while avoiding F subset T.
+
+Moser isotopy (Section 7.1):
+ Given a smooth path of symplectic forms omega_t = omega_0 +
+ d alpha_t, integrate the vector fields X_t defined by
+ iota_{X_t} omega_t = -alpha_t into an isotopy psi_t with
+ psi_t^* omega_t = omega_0. The engine behind "symplectically
+ equalize two forms in the same cohomology class."
+
+Relative Kähler Moser:
+ The version of Moser that preserves a list of complex
+ submanifolds as sets. Uses a normal first-jet correction
+ along each S_j.
+
+Kähler cut (symplectic cut):
+ Take V x C, act by the standard diagonal circle, reduce at
+ a circle-invariant level p_1 + pi |v|^2 = c. The result
+ is a symplectic quotient. Used in Case 3 to build the
+ critical projective bundle Y over the two-point blowup of
+ PP^2.
+
+Complete intersection of quadrics:
+ The set { Q_1 = ... = Q_{n-1} = 0 } in PP^n for n-1
+ general homogeneous quadrics. Smooth of degree 2^{n-1},
+ genus 1 + (n-3) 2^{n-2}, normal bundle
+ O_C(2)^{oplus (n-1)}. This is the "handle" for Cases 1
+ and 2.
+
+Shell:
+ { z in C^n : a < pi sum |z_j|^2 < 1 }. Symplectically
+ equal to Bl_o PP^n minus (E union H_infty) for suitable
+ standard classes. The target of the large-ball reduction.
+
+
+12. A SUGGESTED READING ORDER FOR THE PAPER
+----------------------------------------------------------------
+
+The technical paper (symplectic_ball_packings_original.txt) is
+organized in this order, with the main files as follows:
+
+ intro.tex statement of Theorem + history + sketch
+ comparison.tex Hamiltonian comparison; rearrangement; disks
+ surface.tex surface packing; handle form; layers; Moser
+ geometry.tex Kähler Moser; normal-cone transfer; explicit
+ bundle form; blowup-complement-is-shell
+ applications.tex Case 1 (all small); Case 2 (n >= 4, large)
+ sixdim.tex Case 3 (n = 3, large); cubic; Kähler cut
+ conclusion.tex sufficiency assembly
+ bibliography.tex references
+
+A good first-pass reading order:
+
+ 1. intro.tex, sections 1-2 (statement + easy necessity).
+ 2. comparison.tex, statement of Lemma cmp:comparison and
+ its fixed-point proof.
+ 3. surface.tex, statement of Lemma surf:packing and Step 5
+ (how the walls of a layer hold a closed ball).
+ 4. geometry.tex, statement of Lemma geo:transfer and Lemma
+ geo:shell ("normal-cone transfer" and
+ "blowup-complement-is-shell").
+ 5. applications.tex, Proposition app:small (Case 1).
+ 6. applications.tex, Proposition app:large (Case 2).
+ 7. sixdim.tex, Proposition six:large (Case 3).
+ 8. conclusion.tex.
+
+Sections 3-4 of the paper play a supporting role (relative
+Kähler Moser, projective transfer). They can be skimmed on a
+first reading.
+
+For a truly thorough reading, pay attention to:
+ - the two "regularity" tricks in the surface packing lemma
+ (Step 2: relative area primitive; Step 3: layer path);
+ - why the concavity of P is preserved under the fixed-point
+ iteration;
+ - why the cubic model in Case 3 needs deg O_C(3) = 9 and
+ not 3.
+
+Each of these is where a sloppy proof would break.
+
+
+13. THE ONE-LINE MORAL
+----------------------------------------------------------------
+
+Symplectic geometry is more than volume. It measures
+2-dimensional signed area on every coordinate plane, and this
+strict structure produces extra obstructions you might not
+expect (Gromov's two-ball rules).
+
+But those extra obstructions are tameable in dimension >= 6.
+Once n >= 3, only volume and the two-ball rule survive, and
+the paper shows they are SUFFICIENT. The construction builds
+surface-plus-toric models inside the target, reshuffles fiber
+coordinates via a Hamiltonian isotopy to produce disjoint
+layers, and transports the resulting packing back to the ball
+via normal-cone degenerations and Moser isotopies.
+
+The final answer, in every dimension n >= 3, for every k, and
+for every choice of positive capacities:
+
+ pack iff (sum R_i^n < R^n) and (all R_i + R_j < R).
+
+Not one condition more. Not one condition less.
+
+ --- end ---