Varney's Law: knowledge saturates, and the curve has a shape

A saturated domain does not stop suddenly. It follows a shape, and the shape is this:

dK/dt = alpha * K * (1 - K / K_max) - beta * K^2

where K is the number of things discovered, K_max is the size of the pool, alpha is a discovery rate and beta is a rate at which pairs of discoveries make each other obsolete.

Two terms, doing two different jobs. The first term is a logistic: growth proportional to what you already have, damped by how much of the pool is filled. The second is a correction that only bites once there is a substantial body of knowledge to contradict itself.

What the shape looks like

Early on, K is small, the damping is negligible, and the first term is roughly exponential. Discovery feels explosive, and it is — there is a real reason the early history of a science reads as a rush.

Then the damping takes over. Growth slows, then flattens. The plateau is not a failure of imagination or a shortage of funding; it is what a finite pool with a fixed access rate looks like from inside.

The fixed point, where discovery and obsolescence balance, is

K* = alpha / (alpha / K_max + beta)

and the ratio K*/K_max is the saturation fraction. The number I have quoted elsewhere — 97.5% in classical domains — is that ratio, estimated per domain from growth curves.

What it was fitted to

The law was fitted to USPTO patent data using nonlinear least squares: annual counts of applications and grants, 1960 to 2020, with cumulative grants as the state variable and a scaled application rate as an input. Against a plain logistic and a Gompertz curve, this two-term version fit the annual increments better.

It is a fit. Alpha and beta are empirical constants with no derivation behind them at that stage of the work. I want to be explicit about that, because for a long time this equation was presented in this programme as though it were a law of nature rather than a curve that happened to describe patent counts.

What happened later

It turns out the form can be derived. If you model researchers as agents picking combinations from a finite pool — each attempt novel with probability proportional to the unused fraction of the pool, plus pairwise obsolescence — the mean-field limit is that equation. The logistic term is preferential attachment on a bounded pool. The quadratic term is contradiction.

That derivation, and the simulation that reproduces the fixed point to a tenth of a per cent, is a later post in this series. The exponents stop being free parameters; only the two rate constants remain empirical, and those have units you can argue about.

Which is the general pattern in this programme: a fitted curve becomes a theorem, and what is left over is a much smaller and more honest list of unknowns.