Heaps' law (Herdan 1960 / Heaps 1978)

As a text grows, its number of distinct words grows too — but sublinearly. The more you have already written, the rarer it is to introduce a genuinely new term. Vocabulary growth decelerates as a power of length.

Formula: V(n) = K * n^b

• n — total number of tokens (running words) so far.
• V(n) — the vocabulary: count of distinct word types.
• K — a constant (tens, corpus-dependent).
• b — the exponent, 0 < b < 1, empirically ~ 0.4-0.6.
• Because b < 1, doubling the text far less than doubles the vocabulary. (Heaps' law is the co-emergent partner of zipf-law: a Zipfian frequency distribution produces Heaps-law type growth.)

Transfer. Novelty per unit of output is a decreasing function of volume — the tenth post on a topic introduces far fewer genuinely new ideas than the first. For an agent generating awareness content at scale this is the formal warning against volume-as-strategy: b < 1 means marginal output is mostly recombination of terms already spent, i.e. slop. To keep V(n) climbing you must inject new context — new evidence, new frames, new experience — not more tokens over the same context. Distinctiveness is not free with length; it must be sourced.

Kin: zipf-law · shannon-entropy · models · slop-is-a-context-deficit · anti-mid