The Evidence Matrix

Domain	h (bits)	n	κ	h·ln2 / √κ	Closes?
Genomic 5,627 genomes · BiosphereCodec	1.61 Shannon entropy transition/transversion bias measured	2.00 Tree topology Gromov δ = 0 measured	1.34 GTDB telescope 250 genomes, Spearman peak measured	0.96	✓
Viral 15 families · RNA + DNA viruses	0.8–2.9 Per-family entropy RNA & DNA polymerases measured	2.00 Tree topology phylogenetic reconstruction measured	0.3–4.1 BiosphereCodec per-family embedding measured	r = 0.996	✓
Linguistic 1,015 languages · 18 families	1.65 Index Diachronica 16,496 sound change rules measured	2.00 Gromov δ = 0 Sarkar H² ≈ H³ measured	1.31 (h·ln2)² telescope: 0.75 (compressed) predicted	1.00	✓
Proteomic 14 Pfam families · BLOSUM62	2.81 BLOSUM62 effective alphabet k_eff ≈ 7 measured	2.03 BiosphereCodec embedding measured	3.80 (h·ln2)² awaiting tree validation predicted	1.00	◯
Neuropixels 39 Steinmetz sessions · SPD(180)	1.04 Volume entropy geodesic ball growth, 2.4s measured	2.03 From h + κ p = 0.59 vs n = 2 implied	0.485 Triangle excess bootstrap CI ± 0.005 measured	1.03	✓
GPT-2 (layer 9) 124M params · autoregressive	0.97 Volume entropy activation covariance SPD measured	2.04 From h + κ implied	0.413 Triangle excess 179 covariance windows measured	1.05	✓
BERT (layer 6) 110M params · bidirectional encoder	0.96 Volume entropy activation covariance SPD measured	2.05 From h + κ implied	0.403 Triangle excess 180 covariance windows measured	1.05	✓
DistilGPT-2 (layer 3) 82M params · distilled autoregressive	1.05 Volume entropy activation covariance SPD measured	2.15 From h + κ implied	0.398 Triangle excess 179 covariance windows measured	1.15	✓
fMRI ABIDE Pitt · 20 subjects · cc200	1.70 Volume entropy 20s windows, 60 ROIs measured	2.72 Whole-brain recurrence n > 2 as predicted implied	0.469 Log-Euclidean triangle excess 20s windows measured	1.72	n > 2*
EEG EEGBCI · 20 subjects · EO/EC	0.97 Derived from d_corr + κ EO > EC (p = 0.04) derived	2.19 Correlation dimension Grassberger-Procaccia on AIRM measured	0.284 AIRM triangle excess 64 sensors, alpha band measured	1.19	◯
ViT-Base (layer 12) 86M params · vision encoder	1.01 Volume entropy activation covariance SPD measured	2.00 From h + κ monotonic L1→L12 convergence implied	0.486 Triangle excess 46 covariance windows measured	1.00	✓

Domain

h (bits)

h·ln2 / √κ

Closes?

Genomic

5,627 genomes · BiosphereCodec

1.61

Shannon entropy
transition/transversion bias

measured

2.00

Tree topology
Gromov δ = 0

measured

1.34

GTDB telescope
250 genomes, Spearman peak

measured

0.96

✓

Viral

15 families · RNA + DNA viruses

0.8–2.9

Per-family entropy
RNA & DNA polymerases

measured

2.00

Tree topology
phylogenetic reconstruction

measured

0.3–4.1

BiosphereCodec
per-family embedding

measured

r = 0.996

✓

Linguistic

1,015 languages · 18 families

1.65

Index Diachronica
16,496 sound change rules

measured

2.00

Gromov δ = 0
Sarkar H² ≈ H³

measured

1.31

(h·ln2)²
telescope: 0.75 (compressed)

predicted

1.00

✓

Proteomic

14 Pfam families · BLOSUM62

2.81

BLOSUM62 effective alphabet
k_eff ≈ 7

measured

2.03

BiosphereCodec
embedding

measured

3.80

(h·ln2)²
awaiting tree validation

predicted

1.00

◯

Neuropixels

39 Steinmetz sessions · SPD(180)

1.04

Volume entropy
geodesic ball growth, 2.4s

measured

2.03

From h + κ
p = 0.59 vs n = 2

implied

0.485

Triangle excess
bootstrap CI ± 0.005

measured

1.03

✓

GPT-2 (layer 9)

124M params · autoregressive

0.97

Volume entropy
activation covariance SPD

measured

2.04

From h + κ

implied

0.413

Triangle excess
179 covariance windows

measured

1.05

✓

BERT (layer 6)

110M params · bidirectional encoder

0.96

Volume entropy
activation covariance SPD

measured

2.05

From h + κ

implied

0.403

Triangle excess
180 covariance windows

measured

1.05

✓

DistilGPT-2 (layer 3)

82M params · distilled autoregressive

1.05

Volume entropy
activation covariance SPD

measured

2.15

From h + κ

implied

0.398

Triangle excess
179 covariance windows

measured

1.15

✓

fMRI

ABIDE Pitt · 20 subjects · cc200

1.70

Volume entropy
20s windows, 60 ROIs

measured

2.72

Whole-brain recurrence
n > 2 as predicted

implied

0.469

Log-Euclidean triangle excess
20s windows

measured

1.72

n > 2*

EEG

EEGBCI · 20 subjects · EO/EC

0.97

Derived from d_corr + κ
EO > EC (p = 0.04)

derived

2.19

Correlation dimension
Grassberger-Procaccia on AIRM

measured

0.284

AIRM triangle excess
64 sensors, alpha band

measured

1.19

◯

ViT-Base (layer 12)

86M params · vision encoder

1.01

Volume entropy
activation covariance SPD

measured

2.00

From h + κ
monotonic L1→L12 convergence

implied

0.486

Triangle excess
46 covariance windows

measured

1.00

✓

Ten domains. One equation. Zero parameters. From RNA viruses to transformers, the ratio h·ln2 / √κ gravitates toward 1 wherever both quantities are independently measured. Viral families span a 5× range in mutation rate yet every one lands on the predicted curve (r = 0.996). The dimension n = 2 is not assumed — it emerges from hierarchical structure. Where connectivity deviates from tree-like (thalamic relay, whole-brain fMRI), n rises above 2 — not as noise, but as the theory correctly predicting the geometry of recurrence.

h is the entropy rate of the information code, measured in bits per symbol. For DNA, it reflects the effective alphabet of accessible mutations (~3 transitions per nucleotide). For neural systems, it is the volume entropy — the exponential growth rate of geodesic balls on the SPD covariance manifold, derived from Manning’s theorem (1979). For language, it is the transition entropy of sound changes.

n is the embedding dimension. For tree-structured data (genomes, languages), n = 2 because trees embed isometrically into the hyperbolic plane (Gromov δ = 0). For neural and AI systems, n is implied from independently measured h and κ via the state equation. The 39-session Neuropixels cohort gives n = 2.03 ± 0.36 (p = 0.59 vs n = 2). All three transformer architectures have layers where n ≈ 2.

κ is the sectional curvature, measured by triangle excess on the data manifold. For genomic data, this is the Poincaré ball embedding; for neural and AI data, it is the SPD covariance manifold with the Log-Euclidean or AIRM metric.

h·ln2 / √κ is the closure test. If the state equation holds with n = 2, this ratio should equal n − 1 = 1. A value within 5% of 1.0 indicates the equation closes.