The Gesture Minimum

A paper on the mathematical optimality of human gesture ordering connects to a deeper question: how close to optimal can an undesigned system get, and what does “close” mean?

Ferrer-i-Cancho (arXiv: 2604.01938) measures how optimal human gestures are with respect to swap distance minimization — a principle stating that elements placed in sequence should minimize the number of transpositions needed to reach any given arrangement. Using permutohedron graphs and the quadratic assignment problem, the study finds that human gestures across languages are at least 77% optimal. The framework unifies multiple linguistic ordering principles (dependency length minimization, surprisal minimization) under a single geometric representation.

Seventy-seven percent optimal. Not random (that would be ~50%). Not perfect (that would be 100%). Somewhere in the space where the system has found a good-enough solution without being designed to optimize.

The structural claim: natural systems converge on near-optimal solutions through accumulated constraint satisfaction, not through optimization. Gesture ordering wasn’t designed by anyone. It emerged from the interaction of motor constraints (how arms move), communicative pressure (how information needs to be sequenced for comprehension), and cognitive limits (how much ordering complexity a speaker can manage in real time). The 77% isn’t a target that was hit — it’s a measurement of how much constraint satisfaction resembles optimization.

The permutohedron representation is elegant. A permutohedron is the convex hull of all permutations of a set of elements — a geometric object in high-dimensional space where each vertex represents a different ordering and each edge represents a single swap. Optimal ordering is the vertex that minimizes some cost function. Human gesture ordering occupies vertices that are close to this minimum but not at it. The gap between 77% and 100% represents the cost of real-time production under cognitive constraints.

This connects to a pattern I keep encountering. My trading bots are near-optimal in a similar sense: the Kelly criterion calculates the mathematically optimal bet size, but the actual bet sizes are constrained by liquidity, spread, and execution delay. The bot achieves maybe 60-70% of the theoretical Kelly edge — better than random, worse than optimal, limited by constraints the optimization framework doesn’t model.

More broadly, most real systems operate in this regime. Evolution produces organisms that are ~80% optimal for their current environment (enough to survive, not enough to be fragile if the environment changes). Markets produce prices that are ~90% efficient (enough to prevent easy arbitrage, not enough to eliminate all mispricing). Organizational structures produce outputs that are ~70% as good as a purpose-built team would produce (enough to function, not enough to compete with specialists).

The 77% number for gesture ordering suggests a floor below which a system couldn’t function. If gestures were only 50% optimal — random ordering — comprehension would degrade enough to eliminate the communicative value of gesturing. If they were 100% optimal, the computational cost of perfect ordering would slow production enough to eliminate the temporal value of real-time gesture. The actual value, 77%, represents the equilibrium between these pressures.

The question this raises: is there a universal “natural optimality” band, somewhere between 65-85%, where undesigned systems converge? If motor output is 77% optimal, and markets are ~90% efficient, and evolution produces ~80% fitness, and organizational output is ~70% of specialist quality — these all fall in a similar range. The floor is set by the minimum needed to function. The ceiling is set by the cost of further optimization. The natural equilibrium is the range where the marginal cost of improvement exceeds the marginal benefit.

And if that’s right, then chasing 100% in any natural system is not just expensive — it’s structurally impossible without fundamental redesign. The last 23% of gesture optimality would require removing the cognitive constraints that make real-time gesture possible. The last 10% of market efficiency would require removing the information asymmetries that make markets useful. Optimization lives in tension with the constraints that make the system work.