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Reference table: byte-level harmonic fraction for 49 languages

Name: Byte-Level Harmonic Fraction for 49 Languages
Creator: Richard Hoekstra
Published: 2026-04
License: https://creativecommons.org/licenses/by/4.0/

Data from the irreversibility depth measurement. Method: Helmholtz-Hodge decomposition on byte-level de Bruijn graphs. Source: Wikipedia (~300k bytes per language).

The harmonic fraction f(D) measures the proportion of byte-level transition energy carried by irreversible cycle currents at context depth D. D* is the extrapolated depth where f vanishes. Sorted by D* (characters).

Language	Code	Order	Morph	Script	bpc	D*_c	D*_b	f(1)	f(2)	f(3)	f(4)	f(5)	Cycles
Chinese	zh	SVO	isolating	cjk	2.4	1.2	3.0	0.969	0.777	0.502	0.457	0.405	1886
Japanese	ja	SOV	agglutinating	cjk	2.5	1.5	3.7	0.974	0.761	0.580	0.472	0.427	844
Hindi	hi	SOV	fusional	devanagari	2.5	1.9	4.8	0.966	0.742	0.638	0.545	0.488	152
Korean	ko	SOV	agglutinating	hangul	2.1	2.0	4.2	0.974	0.764	0.569	0.510	0.464	721
Ukrainian	uk	SVO	fusional	cyrillic	1.7	2.2	3.8	0.970	0.722	0.561	0.482	0.435	360
Russian	ru	SVO	fusional	cyrillic	1.7	2.3	3.9	0.974	0.725	0.570	0.490	0.441	435
Mongolian	mn	SOV	agglutinating	cyrillic	1.7	2.3	4.0	0.974	0.731	0.585	0.500	0.450	210
Greek	el	SVO	fusional	other	1.7	2.3	4.0	0.972	0.720	0.577	0.496	0.455	382
Arabic	ar	VSO	fusional	arabic	1.7	2.3	4.1	0.967	0.744	0.630	0.503	0.442	140
Hebrew	he	SVO	fusional	other	1.7	2.4	4.0	0.965	0.742	0.602	0.502	0.433	204
Bulgarian	bg	SVO	fusional	cyrillic	1.7	2.7	4.5	0.971	0.739	0.593	0.521	0.476	337
Serbian	sr	free	fusional	cyrillic	1.5	3.0	4.4	0.972	0.730	0.582	0.516	0.478	530
Czech	cs	free	fusional	latin	1.1	3.3	3.6	0.957	0.712	0.540	0.476	0.448	1194
Slovak	sk	free	fusional	latin	1.1	3.5	3.8	0.958	0.707	0.548	0.484	0.457	1006
Hungarian	hu	SOV	agglutinating	latin	1.1	3.5	3.8	0.953	0.713	0.549	0.491	0.466	1122
Turkish	tr	SOV	agglutinating	latin	1.1	3.8	4.0	0.956	0.751	0.566	0.501	0.474	854
Polish	pl	free	fusional	latin	1.1	3.9	4.0	0.960	0.739	0.565	0.502	0.469	867
Croatian	hr	free	fusional	latin	1.0	3.9	4.0	0.954	0.749	0.568	0.499	0.463	873
Vietnamese	vi	SVO	isolating	latin	1.2	4.0	4.9	0.971	0.721	0.569	0.522	0.496	290
German	de	SOV	fusional	latin	1.0	4.2	4.3	0.951	0.742	0.580	0.510	0.472	1111
Latvian	lv	free	fusional	latin	1.1	4.2	4.5	0.961	0.750	0.586	0.519	0.485	807
Spanish	es	SVO	fusional	latin	1.0	4.2	4.3	0.960	0.725	0.565	0.509	0.477	674
Lithuanian	lt	free	fusional	latin	1.1	4.3	4.6	0.958	0.729	0.576	0.521	0.485	713
Portuguese	pt	SVO	fusional	latin	1.0	4.3	4.4	0.960	0.736	0.573	0.515	0.481	765
Italian	it	SVO	fusional	latin	1.0	4.5	4.5	0.958	0.744	0.571	0.515	0.487	858
Danish	da	SVO	fusional	latin	1.0	4.5	4.6	0.954	0.752	0.586	0.522	0.486	881
Estonian	et	SVO	agglutinating	latin	1.0	4.6	4.7	0.959	0.726	0.568	0.519	0.492	918
French	fr	SVO	fusional	latin	1.0	4.6	4.8	0.960	0.758	0.597	0.530	0.491	802
Romanian	ro	SVO	fusional	latin	1.0	4.6	4.8	0.962	0.729	0.575	0.523	0.495	726
English	en	SVO	fusional	latin	1.0	4.7	4.8	0.957	0.761	0.588	0.523	0.492	702
Dutch	nl	SVO	fusional	latin	1.0	>5	>5	0.956	0.763	0.601	0.543	0.511	667
Swedish	sv	SVO	fusional	latin	1.0	>5	>5	0.963	0.750	0.614	0.568	0.546	451
Norwegian	no	SVO	fusional	latin	1.0	>5	>5	0.960	0.740	0.593	0.543	0.517	620
Indonesian	id	SVO	agglutinating	latin	1.0	>5	>5	0.956	0.772	0.613	0.552	0.528	497
Malay	ms	SVO	agglutinating	latin	1.0	>5	>5	0.964	0.762	0.610	0.556	0.533	575
Thai	th	SVO	isolating	other	2.5	>5	>5	0.969	0.766	0.644	0.579	0.548	314
Swahili	sw	SVO	agglutinating	latin	1.0	>5	>5	0.961	0.779	0.621	0.561	0.533	431
Bengali	bn	SOV	fusional	other	2.5	>5	>5	0.970	0.749	0.629	0.555	0.508	213
Persian	fa	SOV	fusional	arabic	1.8	>5	>5	0.965	0.762	0.659	0.570	0.504	136
Finnish	fi	SVO	agglutinating	latin	1.0	>5	>5	0.961	0.733	0.590	0.546	0.515	908
Georgian	ka	SOV	agglutinating	other	2.3	>5	>5	0.969	0.778	0.672	0.614	0.577	186
Tamil	ta	SOV	agglutinating	other	2.6	>5	>5	0.965	0.780	0.697	0.638	0.598	110
Telugu	te	SOV	agglutinating	other	2.6	>5	>5	0.962	0.780	0.672	0.579	0.518	92
Burmese	my	SOV	isolating	other	2.6	>5	>5	0.970	0.779	0.669	0.610	0.574	159
Uzbek	uz	SOV	agglutinating	latin	1.0	>5	>5	0.970	0.754	0.599	0.546	0.521	750
Irish	ga	VSO	fusional	latin	1.1	>5	>5	0.961	0.735	0.587	0.533	0.509	394
Welsh	cy	VSO	fusional	latin	1.0	>5	>5	0.958	0.742	0.593	0.537	0.511	782
Tagalog	tl	VSO	agglutinating	latin	1.0	>5	>5	0.960	0.767	0.600	0.540	0.509	855
Latin	la	free	fusional	latin	1.0	>5	>5	0.961	0.749	0.601	0.559	0.537	664

Column definitions

bpc	Mean bytes per Unicode character
*D_c**	Irreversibility depth in characters (D*_bytes / bpc)
*D_b**	Irreversibility depth in bytes
f(D)	Harmonic fraction at context depth D = \|\|A_harm\|\|² / \|\|A\|\|²
Cycles	Number of depth-3 harmonic cycles (trigram rotations)

Cite this data

@misc{hoekstra2026atlas,
  author = {Hoekstra, Richard},
  title = {Byte-Level Harmonic Fraction for 49 Languages},
  year = {2026},
  url = {https://richardhoekstra.nl/atlas/reference-table.html}
}