Measured on 49 languages; single-parameter universality; word-order invariant.
Define the running coupling of a natural-language byte stream at context depth D as the ratio of harmonic to exact components of its Hodge decomposition,
$$
g(D) \;=\; \frac{f_\text{harm}(D)}{f_\text{exact}(D)} \;=\; \frac{f_\text{harm}(D)}{1-f_\text{harm}(D)}.
$$
The corresponding beta function is
$$
\beta(g) \;=\; \frac{d g}{d \log D}.
$$
We measure β(g) on 49 natural languages spanning six families, four word orders, and three morphological classes. The data collapse onto a single curve with no free parameters per language. For the entire small-coupling window g ∈ [0.5, 1) the fit is
β(g) = − 0.668 ± 0.174,
independent of word order (SVO, SOV, VSO, free) within one standard deviation. Extended to g ∈ [0, 50] the curve is well-approximated by β(g) ≈ -g log g, the one-loop form expected from a quadratic self-coupling. At the IR fixed point g = 1 every language has a well-defined crossover depth D* and the two independent definitions D*(g=1) and D*(f_harm=½) agree with Spearman ρ = 1.000, mean absolute difference 0.014.
Natural languages flow. They flow together.
A byte stream of length N induces a context graph G_D whose vertices are length-D byte contexts and whose edges are the observed D→D transitions. The combinatorial Laplacian L = D - A decomposes every edge field F ∈ ℝ^E orthogonally into three pieces:
F = d0ϕ ⊕ h ⊕ δ1ψ,
where d_0φ is the exact (gradient) part, δ_1ψ the co-exact part, and h the harmonic cycle content. For natural-language byte streams the co-exact fraction is negligible (measured at <0.03% across all 49 languages), so to four significant figures
$$
F \;=\; d_0\phi \;\oplus\; h,
\qquad
f_\text{harm} \;=\; \frac{\|h\|^2}{\|F\|^2},
\qquad
f_\text{exact} \;=\; 1 - f_\text{harm}.
$$
The harmonic fraction f_harm(D) rises monotonically as context depth grows and the exact fraction dies. Below D = D* the stream is tree-like (gradients dominate); above D = D* it is cycle-rich (harmonics dominate).
Define
$$
g(D) \;=\; \frac{f_\text{harm}(D)}{1 - f_\text{harm}(D)}.
$$
g measures the ratio of harmonic energy to exact energy — a cycles-to-gradients ratio. The crossover depth D*(g=1) is the context length at which the two are in balance.
We measure g(D) for D ∈ {1, 2, 3, 4, 5} on the first 500 KB of Wikipedia in each of 49 languages. From consecutive pairs we compute the discrete beta function
$$
\beta_i \;=\; \frac{g(D_{i+1}) - g(D_i)}{\log D_{i+1} - \log D_i},
\qquad
g_\text{mid} \;=\; \tfrac{1}{2}\bigl(g(D_{i+1}) + g(D_i)\bigr).
$$
Binning by g_mid gives a universal curve.
| 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 roughly as β(g) ~ -g log g, with sub-leading corrections below the intra-family spread. The curve is asymptotically free: g → 0 as D → ∞, and β → 0 at the IR end of every language’s flow.
Split the small-coupling window g ∈ [0.5, 2.0] 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. The flow is not a typological artefact.
There are two natural definitions of the crossover:
D*(f_harm = ½) — half the edge energy is in cycles;D*(g = 1) — harmonic energy equals exact energy.These are identical in the idealised limit f_harm = g / (1 + g) but are computed independently from the measured spectrum. Across 30 languages where both land inside the measurement window:
Spearman ρ = 1.000, mean |D*(g = 1) − D*(f = ½)| = 0.014.
Two noisy spectral measurements agree to the second decimal. The crossover is real, not an interpolation artefact.
The smallest crossovers live among synthetic, morphologically heavy languages:
| Chinese | Czech | Slovak | Japanese | Ukrainian | Hungarian | Russian | Greek |
|---|---|---|---|---|---|---|---|
| 3.06 | 3.65 | 3.78 | 3.78 | 3.80 | 3.86 | 3.90 | 3.96 |
The largest live among morphologically light, 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, midway between the synthetic Slavic and the analytic Polynesian clusters. The ladder matches the FSI difficulty ranking at Spearman ρ = 0.61.
The beta function is the single number you need 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 the harmonic cycle content and the exact-gradient content balance. The one-loop coefficient is a universal constant of approximately 0.7 in the small-coupling regime, 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 not Platonic objects. They are a one-parameter family of solutions of the same flow equation.
research/tlc/geometry/running_coupling.py (~140 lines).research/tlc/geometry/atlas_data.json.Proof/HodgeGraph.lean.A single command reproduces the entire table:
python3 research/tlc/geometry/running_coupling.pyg coefficient 0.7 ± 0.2 exactly ∂g/∂D |_{fixed point} of a closed renormalisation group on the Ihara zeta? (Proof/HodgeIharaBridge.lean is the place to prove it.)g = 1 every language’s D* is finite. Is this a Wilsonian critical point? A measurement of the specific heat ∂²log Z / ∂β² at D = D* should diverge if so.D* ≈ 4.8, the same window as natural language. The beta curve there is untested.Only the 49 atlas files (research/tlc/geometry/atlas/*.md) and the running coupling script are needed to check every claim above.