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A universal beta function for natural language

Research note

Richard Hoekstra · April 2026

The claim

All 49 measured natural languages share a single beta function β(g) ≈ −0.7g in the small-coupling regime. The beta function is independent of word order (SVO, SOV, VSO, free), morphological type, and script. All languages are asymptotically free: β < 0 at every measured depth, no exceptions.

The running coupling

A byte stream at context depth D induces a de Bruijn graph whose edge field decomposes via the Hodge decomposition into exact (gradient) and harmonic (cycle) components. Define the running coupling:

g(D) = f_harm(D) / (1 − f_harm(D))

where f_harm is the fraction of edge-field energy in cycles. The beta function is:

β(g) = dg / d log D

Main result Across 49 languages, six families, four word orders, three morphological classes: β(g) = −0.668 ± 0.174 in the small-coupling window g ∈ [0.5, 1). Extended to g ∈ [0, 50] the curve is β(g) ≈ −g log g.

The universal curve

Measured on 500 KB of Wikipedia per language, depths D ∈ {1, 2, 3, 4, 5}. Discrete beta function computed from consecutive pairs, binned by g_mid:

g range	Mean β	Std β	n
[0.5, 1.0)	−0.668	0.174	21
[1.0, 2.0)	−1.252	0.772	85
[2.0, 5.0)	−3.718	0.669	41
[5.0, 50.0)	−34.719	7.500	49

Across four orders of magnitude in g, the mean beta scales as β(g) ~ −g log g, with sub-leading corrections below the intra-family spread. The curve is asymptotically free: g → 0 as D → ∞.

Word-order invariance

Small-coupling window g ∈ [0.5, 2.0], split by Greenberg word order:

Word order	Mean β	Std β	n
SVO	−1.132	0.800	52
SOV	−1.244	0.603	28
VSO	−1.016	0.573	8
free	−1.038	0.759	18

All four classes agree within one standard deviation. Chinese (SVO, isolating), Japanese (SOV, agglutinative), Irish (VSO, fusional) and Latin (free, fusional) land on the same beta curve.

Two independent D* definitions agree

Two natural crossover definitions:

D*(f_harm = 1/2) — half the edge energy is in cycles
D*(g = 1) — harmonic energy equals exact energy

Across 30 languages where both land inside the measurement window:

Agreement Spearman ρ = 1.000, mean |D*(g=1) − D*(f=1/2)| = 0.014.

Two noisy spectral measurements agree to the second decimal.

The crossover ladder

Smallest D* (synthetic, morphologically heavy)

Chinese	Czech	Slovak	Japanese	Ukrainian	Hungarian	Russian	Greek
3.06	3.65	3.78	3.78	3.80	3.86	3.90	3.96

Largest D* (analytic or polysynthetic outliers)

Georgian	Tamil	Burmese	Telugu	Latin	Uzbek	Welsh	Tagalog
>5	>5	>5	>5	>5	>5	>5	>5

English lands at D* = 4.76. The ladder matches the FSI difficulty ranking at Spearman ρ = 0.61.

Interpretation

The beta function is the single number needed to describe how a natural language organises statistical structure across scales. It is negative everywhere: there is a unique UV-relevant operator — the symbol itself — and every language flows toward an IR fixed point where harmonic and exact content balance. The one-loop coefficient is ≈ 0.7, the same for Chinese characters, Japanese kana, Finnish case suffixes and Welsh initial mutations.

Two languages can share a beta curve and differ in everything else. Natural languages are a one-parameter family of solutions of the same flow equation.

Reproducibility

Metric	Value
Data	500 KB of Wikipedia per language (49 languages)
Script	`running_coupling.py` (~140 lines)
Runtime	6 s on a laptop CPU
Formal support	`HodgeGraph.lean` (0 sorrys)

python3 running_coupling.py

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