Vector Balancing via Directional Total Variation

Abstract: Our main result is a $3\sqrt{2π}$ bound for the Komlós signing problem: every finite family of real vectors of Euclidean norm at most one admits a signed sum of $\ell_\infty$-norm less than this constant, independently of the dimension and the family size. For any $κ\ge0$, if a bounded open convex set supports a probability density with directional total variation at most $κ$ in every unit direction, then its open-set Banaszczyk transform supports another such density with the same $κ$, provided the translation vector $v$ satisfies $κ|v|_2\le1/3$. As a consequence, every finite set system in which each element belongs to at most $t$ sets, where $t\ge1$ is an integer, admits a two-coloring whose imbalance in each set is less than $3\sqrt{2πt}$. This gives the square-root dependence predicted by the Beck-Fiala conjecture. The proof was discovered by the Odin Automatic AI Research Agent.

• The paper establishes an explicit dimension-free bound for the Komlós vector-signing problem, proving the existence of signs such that the maximum absolute value of the signed sum is less then 3√2π.
• The method introduces a stability theory for probability densities controlled by directional total variation, replacing the Gaussian-measure formulation with an invariant preserved under a suitable open-set version of the Banaszczyk transform.
• The result yields an existential discrepancy guarantee for Beck-Fiala bounds, showing a conjecture square-root dependency on maximum element degree, which is qualitatively stronger than classical linear estimates.

Main result and positioning

The paper establishes an explicit dimension-free bound for the Komlós vector-signing problem. Given vectors $v_1,\ldots,v_n\in\mathbb{R}^m$ satisfying $\|v_j\|_2\le 1$, it proves the existence of signs $\varepsilon_j\in\{-1,1\}$ such that

$$\left\|\sum_{j=1}^n \varepsilon_j v_j\right\|_\infty <3\sqrt{2\pi}.$$

Thus the Komlós conjecture is resolved with the explicit universal constant $3\sqrt{2\pi}\approx 7.52$ (2609.11189). The result is existential and does not claim optimality of the constant. It also does not provide a polynomial-time procedure for constructing the signs.

The argument introduces a stability theory for probability densities controlled by directional total variation. This replaces the Gaussian-measure formulation traditionally associated with Banaszczyk’s vector-balancing theorem with an invariant that is preserved under a suitable open-set version of the Banaszczyk transform. The central analytic statement is that if a bounded open convex set supports a density whose directional variations are uniformly bounded by $\kappa$, then the transformed set supports another density with the same bound, provided the translation vector $v$ obeys

$\kappa\|v\|_2\le \frac13.$

The density is allowed to change at every iteration. This feature is essential: the proof does not attempt to preserve one fixed measure under the entire sequence of transformations.

Consequence for discrepancy and Beck–Fiala

The Komlós theorem immediately yields a Beck–Fiala bound. If $A\in\{0,1\}^{m\times n}$ has at most $t$ nonzero entries in every column, then each column of $\|v_j\|_2\le 1$0 has Euclidean norm at most one. Applying the vector-balancing theorem and rescaling gives

$\|v_j\|_2\le 1$1

This establishes the conjectured square-root dependence on the maximum element degree for every positive integer $\|v_j\|_2\le 1$2, with no restriction involving $\|v_j\|_2\le 1$3 (2609.11189). The result is therefore qualitatively stronger than the classical linear estimate $\|v_j\|_2\le 1$4 and removes the logarithmic sparsity condition appearing in some recent asymptotic Beck–Fiala results.

The implication is specifically an existential discrepancy guarantee. It does not establish an efficient coloring algorithm, online prefix control, or concentration for a distribution over colorings. These distinctions matter because previous algorithmic approaches, including the Gram–Schmidt walk, provide substantially different forms of control, while the present proof targets only one final signed sum.

Directional total variation as the invariant

For an integrable function $\|v_j\|_2\le 1$5 and direction $\|v_j\|_2\le 1$6, the paper defines the distributional directional variation

$\|v_j\|_2\le 1$7

For a probability density $\|v_j\|_2\le 1$8 supported in an open convex body $\|v_j\|_2\le 1$9, the admissibility condition is

$\varepsilon_j\in\{-1,1\}$0

This condition is stronger than controlling coordinate-wise derivatives. It imposes a dimension-independent bound simultaneously in every direction. The use of global BV variation is also important for nonsmooth densities: boundary jumps contribute to $\varepsilon_j\in\{-1,1\}$1 even when the density is constant in the interior of its support.

The variation bound has a direct translation interpretation. For $\varepsilon_j\in\{-1,1\}$2,

$\varepsilon_j\in\{-1,1\}$3

Consequently, small directional variation implies that short translations preserve substantial overlap between the density and its translate. The proof exploits this overlap after lifting the problem by one dimension and symmetrically rearranging vertical fibers.

The paper’s stability proposition can be summarized as follows:

If $\varepsilon_j\in\{-1,1\}$4 supports a density with directional variation at most $\varepsilon_j\in\{-1,1\}$5, then the open transform $\varepsilon_j\in\{-1,1\}$6 supports a density with the same variation bound whenever $\varepsilon_j\in\{-1,1\}$7.

The geometric containment

$\varepsilon_j\in\{-1,1\}$8

is what permits sign recovery. Iterating the transform over $\varepsilon_j\in\{-1,1\}$9 produces nested symmetric convex domains. At the final stage, symmetry ensures that the origin lies in the final domain. Backward induction through the containment relation then selects a sign for each vector and yields a point in the original target body.

The initial cube density

The initial admissible body is the cube

$$\left\|\sum_{j=1}^n \varepsilon_j v_j\right\|_\infty <3\sqrt{2\pi}.$$0

The associated density is the product cosine-squared density

$$\left\|\sum_{j=1}^n \varepsilon_j v_j\right\|_\infty <3\sqrt{2\pi}.$$1

The paper proves

$$\left\|\sum_{j=1}^n \varepsilon_j v_j\right\|_\infty <3\sqrt{2\pi}.$$2

Taking $$\left\|\sum_{j=1}^n \varepsilon_j v_j\right\|_\infty <3\sqrt{2\pi}.$$3 makes the variation parameter $$\left\|\sum_{j=1}^n \varepsilon_j v_j\right\|_\infty <3\sqrt{2\pi}.$$4, exactly matching the stability threshold for vectors of norm at most one.

The estimate is obtained by transforming each coordinate through

$$\left\|\sum_{j=1}^n \varepsilon_j v_j\right\|_\infty <3\sqrt{2\pi}.$$5

which yields independent variables with density proportional to $$\left\|\sum_{j=1}^n \varepsilon_j v_j\right\|_\infty <3\sqrt{2\pi}.$$6. These variables admit a Gaussian-scale-mixture representation involving standard Gaussian variables and independent $$\left\|\sum_{j=1}^n \varepsilon_j v_j\right\|_\infty <3\sqrt{2\pi}.$$7 variables. Conditioning on the scale variables and applying Jensen’s inequality improves the elementary Cauchy–Schwarz estimate from $$\left\|\sum_{j=1}^n \varepsilon_j v_j\right\|_\infty <3\sqrt{2\pi}.$$8 to $$\left\|\sum_{j=1}^n \varepsilon_j v_j\right\|_\infty <3\sqrt{2\pi}.$$9.

The paper further shows that $3\sqrt{2\pi}\approx 7.52$0 is asymptotically sharp for this particular product-density family as the dimension tends to infinity. Specifically, for the diagonal direction $3\sqrt{2\pi}\approx 7.52$1,

$3\sqrt{2\pi}\approx 7.52$2

This does not imply that the cube constant or the Komlós constant is optimal among all possible densities or target bodies. It only identifies the limiting worst-direction behavior of the chosen product construction.

The prescribed-section principle

The technically central part of the paper is a mechanism for passing from averaged directional bounds in a lifted body to a density on one prescribed section.

Let $3\sqrt{2\pi}\approx 7.52$3 be a bounded open convex set symmetric in the vertical coordinate, and write

$3\sqrt{2\pi}\approx 7.52$4

Suppose a probability density $3\sqrt{2\pi}\approx 7.52$5 supported in $3\sqrt{2\pi}\approx 7.52$6 has uniformly bounded horizontal variations,

$3\sqrt{2\pi}\approx 7.52$7

and has mean absolute height at least $3\sqrt{2\pi}\approx 7.52$8. The prescribed-section lemma shows that $3\sqrt{2\pi}\approx 7.52$9 supports a density $\kappa$0 satisfying the full directional bounds

$\kappa$1

The proof proceeds through anisotropic Cheeger energies. For a finite mixture of directions,

$\kappa$2

the corresponding energy is

$\kappa$3

The associated Cheeger value $\kappa$4 is the infimum of this energy over probability densities supported in $\kappa$5. The paper proves that $\kappa$6 is convex under Minkowski interpolation of convex domains:

$\kappa$7

The proof regularizes the possibly degenerate integrand $\kappa$8 by smooth strongly convex norms, invokes Brunn–Minkowski-type convexity for anisotropic $\kappa$9-Laplace eigenvalues, and passes to the limit $v$0. The limiting identity identifies the first anisotropic eigenvalue with the Cheeger constant.

A separation argument then converts bounds on every finite weighted combination of directional variations into the existence of one density satisfying all individual directional bounds. This step avoids any need for simultaneous attainment of infinitely many variational problems. Compactness in $v$1 and lower semicontinuity of BV variation complete the passage from finite direction sets to all directions.

The implication is that the density extracted from a section need not be one of the original vertical slices. It is reconstructed globally from the family of slice-wise variational bounds.

Lift, rearrangement, and the stability threshold

For a convex body $v$2 and vector $v$3, the paper considers the lift

$v$4

The vertical fiber above $v$5 consists of those $v$6 for which the line $v$7 remains in $v$8. Symmetric fiber rearrangement produces a convex body $v$9 whose height-one section is exactly $\kappa\|v\|_2\le \frac13.$0.

Starting from an admissible density $\kappa\|v\|_2\le \frac13.$1 on $\kappa\|v\|_2\le \frac13.$2, the lifted density is

$\kappa\|v\|_2\le \frac13.$3

The paper rearranges each vertical fiber symmetrically and decreasingly. This rearrangement preserves total mass, preserves support inside the symmetrized lift, and contracts horizontal $\kappa\|v\|_2\le \frac13.$4 distances. Hence it cannot increase horizontal directional variation.

The key quantitative identity controls the retained vertical height:

$\kappa\|v\|_2\le \frac13.$5

Using the translation inequality gives

$\kappa\|v\|_2\le \frac13.$6

If $\kappa\|v\|_2\le \frac13.$7 and $\kappa\|v\|_2\le \frac13.$8, then

$\kappa\|v\|_2\le \frac13.$9

The prescribed-section principle can therefore be applied at height one, yielding an admissible density on $A\in\{0,1\}^{m\times n}$0. The constant $A\in\{0,1\}^{m\times n}$1 is thus not inserted arbitrarily: it is the threshold that ensures the rearranged density retains sufficient average height for the height-one section to remain accessible.

The truncation height $A\in\{0,1\}^{m\times n}$2 is also structurally constrained by the sign-recovery containment. Larger truncation intervals can improve the height calculation in isolation, but they generally destroy the inclusion $A\in\{0,1\}^{m\times n}$3. Within this particular lift-and-rearrangement scheme, the choice of $A\in\{0,1\}^{m\times n}$4 balances the geometric and analytic requirements.

Relation to prior discrepancy methods

The paper differs from Gaussian-measure approaches in several respects. Banaszczyk’s theorem reduces vector balancing to placing vectors inside convex bodies of sufficiently large Gaussian measure, but fixed cubes have Gaussian measure that deteriorates with dimension. The present proof instead constructs a density supported directly on a cube and controls its directional BV variation.

It also differs from partial-coloring arguments. Partial coloring controls a positive fraction of variables at each stage, but under column-norm hypotheses progress in the number of fixed variables does not directly control every row. The directional-variation transform avoids this issue by processing every vector through a full-signing recursion.

The result is numerically stronger than the previously cited $A\in\{0,1\}^{m\times n}$5 Komlós bound (Bansal et al., 5 Aug 2025, Bansal et al., 28 Aug 2026) and the earlier $A\in\{0,1\}^{m\times n}$6 estimates, because it is independent of both dimension and family size. It is nevertheless not algorithmically stronger: the construction of the recursive domains, the membership queries, the density-selection steps, and the backward sign recovery are not shown to be computationally efficient.

Limitations and open questions

The paper explicitly leaves several issues unresolved. First, the constant $A\in\{0,1\}^{m\times n}$7 is not claimed to be optimal. The lower bound $A\in\{0,1\}^{m\times n}$8 for the Komlós constant remains compatible with a substantial gap (Kunisky, 2021). Even within the present method, optimality of the product cosine-squared density does not imply optimality of the initial cube construction.

Second, the proof is nonconstructive in its present form. The separation and compactness arguments establish existence of densities, but no polynomial-time representation or oracle access model is supplied. In particular, it is open whether the directional-variation framework can be converted into an efficient signing algorithm with a comparable constant.

Third, the theorem controls only the final signed sum. It does not yield analogous bounds for every prefix of an ordering, unlike online discrepancy results such as those studied in (Smirnov et al., 13 May 2026) and (Altschuler et al., 15 Jul 2026). It also does not produce a randomized signing distribution with sub-Gaussian discrepancy tails.

Finally, the threshold $A\in\{0,1\}^{m\times n}$9 is optimal only for the specific truncation, linear translation estimate, and rearrangement argument used here. The paper does not determine the optimal stability threshold for alternative lifts, densities, or rearrangement schemes.

Conclusion

The paper proves a dimension- and cardinality-independent Komlós bound

$t$0

by introducing directional total variation as a stable invariant under an open Banaszczyk transform. Its main technical contributions are the convexity of anisotropic Cheeger costs under Minkowski interpolation, the extraction of a common density from finite directional bounds, and the lift–rearrangement argument that preserves the invariant through each signing step. The resulting Beck–Fiala estimate $t$1 holds for all degrees $t$2, but the construction remains existential and leaves both constant optimization and algorithmic realization open (2609.11189).