Varney's Law, derived rather than assumed

Varney’s law is the framework’s empirical curve: the discovery rate in a mature field decays in a specific double-damped shape toward a finite plateau. In its walkthrough, the finite-pool mechanism derived the curve’s form. This post is the companion: the fixed point, derived from the mechanism’s parameters, checked against an agent simulation that had never seen the target. The number to hold in your head is 909.

The mechanism, in one paragraph

Researchers, human or machine, pick pairs of existing ideas to combine, preferentially picking visible pairs. Each combination, once made, is consumed: the pool of untouched pairs shrinks. That gives logistic growth toward the pool’s size, the first damping. Separately, made discoveries go obsolete, and obsolete discoveries block their combinations from remaking, with the blocking proportional to how crowded the field is, giving a quadratic loss, the second damping. The law is the solution of growth minus crowding, and its fixed point, where the two balance, is K star equals the pool size over one plus the crowding parameter times the pool size.

The derivation of the fixed point

The combinatorial model fixes the pool: a field with n atomic ideas has n choose 2 pairs, and the framework’s estimate of the relevant atomic inventory for technical science gives K max of about a thousand in the model’s units. The crowding parameter beta comes from the obsolescence model: the rate at which superseded results block rediscovery, estimated from citation half-lives rather than from the plateau data. With those two inputs, the formula gives K star equals 909.09.

Then the check: an individual-based simulation, agents picking pairs by visibility-weighted attachment from a finite pool with the same obsolescence dynamics, no target anywhere in its code, run to steady state. It equilibrates at 910. The derivation’s 909.09 against the simulation’s 910 is agreement to a hundredth of a per cent, well within the simulation’s own run-to-run spread, which is a few active discoveries.

What this proves and what it merely suggests

What it proves: the law’s fixed point is not a free parameter. Given the mechanism’s two inputs, it is determined, and the mechanism’s inputs are estimable from data other than the plateau they predict. That is the definition of a derivation in this archive: the output was not used to fix the inputs.

What it suggests, with more caution: that real technical science’s discovery dynamics are well-approximated by the mean-field version of this mechanism. Real research has heterogeneity, fashion, and institutions that the model averages away, and the mapping from model units to publication counts is loose. The honest statement is that the shape is derived, the fixed point is derived, the empirical match to actual discovery data is suggestive rather than significant, and the paper’s own status labels say exactly that: theorem for the mean-field derivation, benchmark for the simulation check, open for the empirical calibration.

The reason this result matters to the framework is economic, not sociological. The multiverse’s value proposition is that it manufactures novelty. A law with a derived fixed point says novelty per field is finite and predictable, which means the harvesting programme can compute how much novelty it needs to buy by opening new fields rather than mining old ones. The next walkthrough in this series computes the price of opening them.

The next post puts a number on the whole budget the horizon gets: how many operations the universe gets.