Symplectic ball packings in higher dimensions

Source and verification

This article explains family 343 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

Symplectic maps preserve volume, but volume alone does not decide whether balls fit inside a ball. This family reports that, in real dimension at least six, precisely one further collection of inequalities is needed: every pair of source capacities must sum to less than the target capacity. The construction combines a rearrangement argument, an exact deformation over a surface with a handle, and geometric models that recover nearly all the target volume. (OpenAI, 2026)

🏷️ Symplectic embeddings and capacity

A symplectic form is a closed, nondegenerate two-form. On Euclidean space with coordinates $(x_1,y_1,\dots ,x_n,y_n)$, the standard example is

$${\omega }_0=\sum _{j=1}^{n}d{x}_j\wedge d{y}_j.$$

An embedding $F$ is symplectic when $F^{\ast }{\omega }_0={\omega }_0$. It consequently preserves the volume form ${\omega }_0^n/n!$. It can substantially deform a round ball; the question concerns symplectic embeddings, rather than arrangements of rigid Euclidean balls.

The manuscript labels a ball by its capacity $R$:

$$\begin{array}{c}{B}^{2n}(R)=\{z\in {\mathbb{C}}^n:c(z)\le R\},\\ c(z)=\pi \sum _j|z_j{|}^2.\end{array}$$

Its Euclidean radius is $\sqrt{R/\pi }$, and its volume is $R^n/n!$. Thus capacity has the units of area. Confusing capacity with radius changes both obstruction formulas.

A map from a closed source ball is required to extend symplectically to an open neighborhood of that ball. The images of the finitely many closed sources must be disjoint compact subsets of the open target. These conventions explain why the theorem uses strict inequalities.

Criterion reported by family 343

Let $n\ge 3$, $k\ge 1$, and let all capacities be positive. The manuscript proves that disjoint neighborhood embeddings of the balls $B^{2n}(R_i)$ into $\mathrm{Int}{B}^{2n}(R)$ exist exactly when

$$\begin{array}{c}\sum _{i=1}^k{R}_i^n<R^n,\\ R_i+R_j<R(i<j).\end{array}$$

For one ball, the second condition is empty. The assertion concerns arbitrary positive real capacities and every finite number of balls. (OpenAI, 2026)

The real dimension assumption is essential: the manuscript does not assert that these conditions suffice in dimension four. Products of four-dimensional balls with a fixed surface are also different packing problems.

🏷️ Necessity and elementary consequences

Compactness places any proposed packing inside a smaller target of capacity $\sigma <R$. Volume preservation gives

$$\sum _i{R}_i^n\le {\sigma }^n<R^n.$$

Gromov’s two-ball obstruction, used as a classical input in the manuscript, gives $R_i+R_j\le \sigma$ for every pair. Passing to $R$ makes that inequality strict too. The difficult direction is sufficiency.

Two capacity-$0.6$ balls in a capacity-one six-dimensional target illustrate the distinction: their combined volume fraction is $2(0.6{)}^3=0.432$, but their capacities sum to $1.2$, so they cannot be packed.

For $k\ge 2$ equal balls of capacity $c$ in a unit target, the criterion becomes

$$c<\min \{k^{-1/n},\frac{1}{2}\}.$$

Their volume fraction is $k{c}^n$. Its supremum is therefore $\min \{1,k/2^n\}$; the strict closed-source convention prevents attainment at the upper endpoint. In six dimensions, nine balls of capacity $0.48$ satisfy both conditions and occupy the exact volume fraction $0.995328$. This is a consequence of the reported criterion, not a numerical construction of an embedding.

Conjugating source and target by the same Euclidean dilation reduces the proof to $R=1$. Although a dilation itself is only conformally symplectic, the two scale factors cancel in the conjugated embedding.

🏷️ Moment coordinates and area profiles

Put $m=n-1$. The construction uses a two-dimensional base surface and a $2m$-dimensional fiber. On that fiber use moment and angle coordinates

$$\begin{array}{c}{p}_j=\pi |z_j{|}^2,\\ {\theta }_j=\arg (z_j)/(2\pi ),\\ {\omega }_V=\sum _{j}d{p}_j\wedge d{\theta }_j.\end{array}$$

Each angle has period one. Polar coordinates are singular at $z_j=0$, but the maps and forms in the proof must be smooth in the original complex coordinates there.

A downward polytope $\Delta$ in the nonnegative moment orthant has the property that lowering any coordinate preserves membership. In the manuscript it is cut out by finitely many inequalities $b_{\nu }\cdot p\le c_{\nu }$ with nonnegative coefficients. Its inverse image under the moment map is a toric domain $V$. A toric function depends only on $p$.

Fix the fiber moments and write $t={\sum }_j{p}_j$. In a ball of capacity $r$, the remaining complex coordinate has available area

$$(r-t{)}_{+},x_{+}=\max \{x,0\}.$$

This is the ball’s area profile. Integration recovers its volume:

$${\int }_{{\mathbb{R}}_{\ge 0}^m}(r-t{)}_{+}dp=\frac{r^{m+1}}{(m+1)!}.$$

To see this, the simplex $\{t\le h\}$ has volume $h^m/m!$, so integration of a function of $t$ has density $t^{m-1}/(m-1)!$. Integrating $(r-t)t^{m-1}$ from zero to $r$ gives the formula.

Let $H(p)$ be the limiting total base area at moment $p$. Choose auxiliary capacities $r_i$ strictly larger than the requested ones. The unused area profile is

$$P(p)=H(p)-\sum _i(r_i-t{)}_{+}.$$

The volume inequality produces ${\int }_{\Delta }Pdp>0$. It does not imply $P(p)>0$ at every moment, so a direct stack of the ball profiles can fail. The next lemma changes the fiber identification to overcome that problem.

🏷️ Hamiltonian comparison from positive mean

A Hamiltonian $Q$ generates a vector field by ${\iota }_{X_Q}{\omega }_V=-dQ$. Its flow preserves the symplectic form. A Hamiltonian diffeomorphism is the endpoint of such a time-dependent flow.

The comparison mechanism

Suppose $P$ is continuous and concave on $\Delta$, has positive mean, and is uniformly positive outside $\tau \Delta$ for some $0<\tau <1$. The manuscript constructs a smooth toric function $K$ and a Hamiltonian diffeomorphism $q$ of $V$ such that

$$(K-P)(qu)<K(u).$$

Near every outer face, the generating Hamiltonians commute with its defining moment. Thus the isotopy respects the boundary constraints. (OpenAI, 2026)

Here an outer face means one defined by $b_{\nu }\cdot p=c_{\nu }$, rather than a coordinate axis. The boundary conclusion is part of the lemma, not an optional refinement.

The analytical ingredient is increasing rearrangement. For a convex function $f$ on $[0,1]$, its increasing rearrangement can be written

$$\begin{array}{c}g(h)=\underset{0\le a\le 1-h}{\min }M(a,h),\\ M(a,h)=\max \{f(a),f(a+h)\}.\end{array}$$

Convexity makes its sublevel sets intervals. The formula selects the least level whose sublevel interval has length at least $h$. It consequently preserves the distribution of values and the integral. Applied successively along moment coordinates, rearrangement gives an operator $\mathcal{R}$ preserving convexity, integrals, order, and addition of constants. It is nonexpansive in uniform norm and produces coordinatewise nondecreasing functions.

For $0<\lambda <1$, consider the contraction

$$F⟼\mathcal{R}(\lambda F-P).$$

It acts on the complete space of continuous convex functions. Its fixed point $K_{\lambda }$ satisfies

$${\overline{K}}_{\lambda }=-\frac{\overline{P}}{1-\lambda }.$$

The proof needs control of the maximum as well as this negative mean. Choose $d>0$ below both $\overline{P}$ and the outer-region lower bound for $P$. If the maximum $M_{\lambda }$ were at least $-d/(1-\lambda )$, then $\lambda K_{\lambda }-P<M_{\lambda }$ on the outer region. On the inner region, convexity gives

$$K_{\lambda }(\tau v)\le \tau M_{\lambda }-\frac{(1-\tau )\overline{P}}{1-\lambda }.$$

For $\lambda$ sufficiently close to one this also forces $\lambda K_{\lambda }-P<M_{\lambda }$ there. Rearrangement cannot raise a maximum, contradicting the fixed-point equation. Hence $(1-\lambda )K_{\lambda }<-d$ everywhere, and

$$\mathcal{R}(K_{\lambda }-P)\le K_{\lambda }-d.$$

This strict gap permits approximation. The manuscript realizes each coordinate rearrangement, up to arbitrarily small one-sided error, by smooth families of planar area-preserving disk maps. Nested disks with prescribed areas are first positioned smoothly; Moser’s method corrects the maps to preserve area. Their Hamiltonian lifts may change the other angles but keep the other moments fixed.

The finitely many approximation errors are chosen to total less than half the gap. The resulting composition gives the required inequality, and a final uniform smoothing of $K_{\lambda }$ retains it. Supports and collars are chosen during the construction so that the Hamiltonians commute with all relevant outer-face moments.

🏷️ A handle converts the gap into disjoint layers

The base is a punctured closed surface of positive genus, exhausted by compact bordered surfaces ${\Sigma }_{\ell }$. The models have forms

$${\Omega }_{\ell }={\omega }_V+d{\Gamma }_{\ell },$$

where ${\Gamma }_{\ell }$ is horizontal—it vanishes on fiber tangent vectors—and its coefficients are toric. If $d_C$ denotes differentiation in the base variables, assume $d_C{\Gamma }_{\ell }>0$ and that its integrated base areas converge uniformly to $H$.

The key positivity identity is

$${\Omega }_{\ell }^{m+1}=(m+1){\omega }_V^m\wedge d_C{\Gamma }_{\ell }.$$

Terms involving two mixed base-fiber factors vanish because they contain too many moment differentials. Thus positivity of the base area guarantees nondegeneracy of the total form.

A handle supplies two transverse annuli and a closed one-form $\beta$ supported around one essential circle, with period one around the other. The construction concentrates most of the available area on one annulus while retaining a small positive background area. Zero-integral changes of area admit compactly supported primitives, so this redistribution gives an exact symplectic deformation.

In the upper part of the annulus the fiber coordinates are changed by $q$, while in the lower part they remain unchanged. Hamiltonian suspension supplies the additional horizontal term needed to make this change symplectic. The comparison inequality is equivalent to positivity of

$$K(q^{-1}v)-K(v)+P(v).$$

That positive residual permits a coefficient along the annulus to increase through successive increments, each strictly larger than one ball profile. The increments occupy disjoint base intervals; the remaining surplus occupies a separate interval. The exact deformation is arranged to match at both ends using the global closed form $\beta$.

After a toric angular shear, each packing layer has the product form ${\omega }_V+dx\wedge dT$ and contains the region

$$0<T<1,0<x<h_i(p),$$

where $h_i(p)>(r_i-t{)}_{+}$. A planar area-preserving map places the last coordinate of a requested ball into this strip with $x<w+\delta$, where $w$ is its radial area and $\delta$ is smaller than the auxiliary capacity slack. For requested capacity $r_i^{\prime }<r_i$, this gives

$$x<w+\delta <r_i-t<h_i(p)$$

on the ball, since $w+t\le r_i^{\prime }$. The inequalities have uniform slack on the compact source, giving an embedding on a neighborhood.

Finally, relative Moser transport returns the layers to the original model. To check the boundary, solve ${\iota }_Y\Omega =-\alpha$ for the variation primitive $\alpha$. Horizontality gives $\alpha (X_{\mu })=0$ for an outer-face moment $\mu$, hence $d\mu (Y)=0$ near that face. The vector field also vanishes near the base boundary. This is why boundary commutation was retained throughout the comparison argument.

Only the limiting profile $P$ needs to be concave. Uniform convergence of finite-model areas preserves the gap; no concavity claim about every finite exhaustion is needed.

🏷️ Models for the three capacity regimes

The geometric part of the manuscript supplies the models and transports their compact packings into the target. Its auxiliary constructions use complex geometry. We record the resulting area profiles and transfer requirements here without developing algebraic geometry.

Capacity regime	Surface model	Required volume limit
All capacities below $1/2$	Positive-genus base in every $n\ge 3$	$1/n!$
One capacity at least $1/2$, $n\ge 4$	Positive-genus model of the exterior shell	$(1-a^n)/n!$
One capacity at least $1/2$, $n=3$	Genus-one base with two marked points	$(1-a^3)/6$

In the first regime, put $m=n-1$ and truncate the simplex at $t\le h<1/2$. The limiting base area is

$$H(p)=2^m(1-2t).$$

Its integral tends to $1/n!$ as $h\uparrow 1/2$. Choose auxiliary capacities below $h$ with total volume still below that integral. Then $P$ is affine minus convex caps, hence concave. Near the outer face all caps vanish, leaving a uniformly positive $H$. These facts verify each hypothesis of the surface packing lemma.

The pairwise inequalities allow at most one requested capacity at least $1/2$. Reserve a slightly larger central ball of capacity $a>1/2$, retaining

$$r_i<1-a,\sum _i{r}_i^n<1-a^n$$

for the remaining auxiliary balls. They will occupy the shell with radial capacity between $a$ and one.

For $n\ge 4$, set $u=p_1$ and $w={\sum }_{j=2}^{n-1}{p}_j$. The shell construction yields

$$H(p)=2^{n-2}(1-u-2w)$$

on truncations $u\le h_1<1-a$ and $w+h_{2}u\le h_2$, with $h_2<1/2$. Its total volume approaches the full shell volume as the truncations approach their limits. Again $P$ is concave, the auxiliary ball simplices stay away from outer faces, and the outer surplus is positive.

🏷️ The six-dimensional shell

In $n=3$, the preceding shell construction would have a genus-zero base, so it cannot supply the required handle. The manuscript replaces it by a genus-one model with two marked points.

Write $s=1-a$ and $b=(2-a)/3$. Its limiting moment domain and profile are

$$\begin{array}{c}0\le p_1\le s,p_2\ge 0,\\ p_1+p_2\le b,\\ H_0(p)=9(b-p_1-p_2)\\ -2(s-p_1-p_2{)}_{+}.\end{array}$$

Each omitted marked point removes a curvature mass $(s-p_1-p_2{)}_{+}$. Locally this comes from a logarithmic term with precisely that coefficient in the reduced potential. Subtracting the two masses from the total curvature gives $H_0$; uniform convergence over compact moment truncations follows from monotone convergence of continuous area functions and Dini’s theorem.

As a function of $t=p_1+p_2$, the slopes of $H_0$ are $-7$ below $s$ and $-9$ above $s$. The slope decreases, so the profile is concave. Direct integration gives

$$\begin{array}{rl}{\int }_{{\Delta }_{\infty }}{H}_{0}dp& =\frac{s}{2}-\frac{s^2}{2}+\frac{s^3}{6}\\ & =\frac{1-a^3}{6}.\end{array}$$

Truncating strictly below $p_1=s$ and $p_1+p_2=b$ retains enough volume for the auxiliary balls. Their cap supports lie in the interior, so the same comparison and surface lemma apply.

A final transfer must preserve the first moment $p_1$ exactly to keep the packing below a prescribed cut level. Equivariance alone preserves a moment only up to a constant. The manuscript fixes that constant by transporting a fixed point on a section where both normalized moments vanish. This permits the symplectic cut and subsequent transport into the shell.

All transfers act on neighborhoods of compact subsets and preserve avoidance of the excluded boundary pieces. Finitely many compositions therefore retain neighborhood embeddings of the closed source balls. Rational auxiliary parameters are chosen inside the strict inequalities; the requested capacities remain arbitrary real numbers.

🏷️ Key steps and proof outlines

A positive integral becomes a uniform rearrangement gap

The fixed-point map $F\mapsto \mathcal{R}(\lambda F-P)$ contracts uniform distance by $\lambda <1$. Integral preservation gives a negative mean for its fixed point, but a negative mean is insufficient for the later pointwise layer separation. The source combines convexity on the inner dilate with positivity of $P$ in the outer collar to bound the maximum as well. If that maximum were too large, both regions would make $\lambda K_{\lambda }-P$ strictly smaller than it, contradicting the fixed-point identity.

The resulting gap $\mathcal{R}(K_{\lambda }-P)\le K_{\lambda }-d$ survives all finite approximation steps. Smooth Hamiltonian realizations are chosen with errors totaling less than $d/2$. Uniform convergence of the integrated finite-surface areas then retains a positive residual on one actual compact model. This explains why the final construction needs only concavity of the limiting profile, not of every approximation. (OpenAI, 2026)

Zero total area gives a compactly supported primitive

The area redistribution over the base must be exact and fixed near its boundary. The local primitive can be written explicitly. For a compactly supported density $h(x,y)$ of integral zero, choose a compactly supported $\rho$ with integral one and set

$$\begin{array}{rl}{h}_0(y)& =\int h(x,y)dx,\\ A(x,y)& ={\int }_{-\infty }^x[h(v,y)-\rho (v)h_0(y)]dv,\\ B(y)& ={\int }_{-\infty }^y{h}_0(v)dv.\end{array}$$

Then $\alpha =Ady-\rho (x)B(y)dx$ is compactly supported and $d\alpha =hdx\wedge dy$. The first subtraction makes the $x$ primitive compactly supported; zero total integral does the same for the $y$ primitive. Since the construction is linear, smooth dependence on moment parameters is preserved.

On a general compact bordered surface, split into coordinate disks, remove each piece’s integral with a bump, and move the compensating bumps through a fixed chain of overlapping disks. Their total coefficient is zero. This produces one primitive with support fixed away from the boundary for the entire parameter family. (OpenAI, 2026)

The Moser vector field respects every face simultaneously

For ${\partial }_{\lambda }{\Omega }_{\lambda }=d{\alpha }_{\lambda }$, solve ${\iota }_Y{\Omega }_{\lambda }=-{\alpha }_{\lambda }$. Near an outer moment face, its standard Hamiltonian field satisfies ${\iota }_{X_{\mu }}{\Omega }_{\lambda }=-d\mu$. Horizontality gives ${\alpha }_{\lambda }(X_{\mu })=0$, and evaluating the Moser equation on that field gives $d\mu (Y)=0$. Trajectories cannot cross the face in either direction.

The same calculation applies to every defining face at once, including corners where several meet. Vanishing of the primitive in a base-boundary collar fixes that collar. Compactness supplies a flow on a neighborhood of the packed region. Thus the deformation transports neighborhood embeddings of closed balls, rather than only maps on their interiors.

Finally the enlarged auxiliary capacities leave strict slack between each requested ball profile and its layer height. That slack pays for cutoffs, planar area maps, and the final transfer into the target. The geometric model constructions supply the stated limiting areas; the analytic gap and boundary calculation explain how those areas become a genuine disjoint packing.

🏷️ Connections

The nearby Lagrangian family uses generating families and Hamiltonian transport to distinguish isotopy classes. Here Hamiltonian transport instead changes how fiber area is distributed, while exact Moser deformation returns the packing to the original symplectic form.

Taming and compatibility explains a separate four-dimensional interaction between positivity and closed forms. The present criterion begins in dimension six and depends on the particular geometry of ball targets.

🏷️ Manuscripts and verification scope

• Symplectic Ball Packings in Higher Dimensions — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The page selects the full packing equivalence and its necessity direction, including the dimension restriction, capacity normalization, strict inequalities, and neighborhood convention for closed balls. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. Symplectic Ball Packings in Higher Dimensions.