D(h, A) = 0.38 − cos(h, A) ← divergence from equilibrium cosine
n̂(h, A) = (h − A) / ‖h − A‖ ← Euclidean radial direction
B(h) = 1 − |cos(h,A_E) − cos(h,A_C)| ← E–C boundary proximity
Three learned anchor vectors A_E, A_C, A_N define the label geometry. The attractor is a ring at cos(h, A_y) = 0.38, not the anchor itself.
At inference all three anchors compete simultaneously — whichever basin has the strongest geometric pull wins.
Force magnitudes are cosine-based; force directions are Euclidean radial. These are geometrically inconsistent (true cosine gradient is tangential). Correct description: discrete-time attractor dynamics with anchor-directed forces. Energy-like, not exact gradient flow.
Most classifiers do: h → linear layer → logits. One step, no dynamics.
Livnium does: h₀ → L steps of geometry-aware state evolution → logits. The final state h_L is dynamically shaped before readout — it isn't just a linear projection of h₀.
The specific things that are different:
1. Classification as attractor dynamics, not a lookup.
The state h moves through space across L steps under anchor forces before the classifier reads it. The label isn't computed from the raw embedding — it's read from where the state settled.
2. The force geometry is deliberately inconsistent — and that's measured.
Force magnitudes follow cosine divergence D(h, A) = 0.38 − cos(h, A). Force directions are Euclidean radial n̂ = (h − A) / ‖h − A‖. These are not the same thing — the true gradient of a cosine energy is tangential on the sphere, not radial. The mean angle between these two directions is 135.2° ± 2.5° (measured, n=1000). This means the system is running explicit physical forces, not gradient descent on the written energy.
3. The attractor is a ring, not a point.
The equilibrium condition is cos(h, A_y) = 0.38, which defines a ring on the hypersphere — not the anchor itself. The system settles to a proximity zone, not a target location. Standard energy minimisation would push to the anchor; this stops at the ring.
4. Proven local contraction.V(h) = (0.38 − cos(h, A_y))² is a Lyapunov function that decreases at every step when δ_θ = 0 (proven analytically, confirmed empirically on 5000 samples). Livnium is a provably locally-contracting pseudo-gradient flow. Most residual classifiers have no such stability guarantee.
5. Inference is a single unsupervised collapse.
Training uses s_y · D(h, A_y) — only the correct anchor pulls. At inference, all three anchors compete with no label. The label is implicit in which basin wins. Cost: 1× forward pass through a small MLP, 428× faster than BERT on CPU.
What it isn't: global convergence is not proven (finite step size + learned residual δ_θ can escape the basin). The geometric inconsistency is not fixed. It isn't yet competitive with fine-tuned transformers on accuracy. Whether iterated attractor dynamics outperform a standard deep residual block at equivalent parameter count is an open question.
Results — SNLI NLI Classification
Accuracy (SNLI dev set)
Class
Accuracy
Overall
76.32%
Entailment
87.5%
Contradiction
81.2%
Neutral
70.9%
Model Config
Parameter
Value
Dim
256
Collapse layers
6
Encoder
Pretrained bag-of-words embeddings (frozen)
Parameters
~2M
Speed vs BERT (CPU, batch size 32)
Model
ms / batch
Samples / sec
Full SNLI train (549k)
Livnium
0.4 ms
85,335 / sec
~6 sec
BERT-base
171 ms
187 / sec
~49 min
428× faster than BERT-base on CPU.
Lyapunov Analysis
Define V(h) = D(h, A_y)² = (0.38 − cos(h, A_y))²
V = 0 at the attractor ring. When δ_θ = 0, V decreases at every step (mean ΔV = −0.00131). Analytically:
∇_h cos · n̂ = −(β · sin²θ) / (α · ‖h − A‖) ≤ 0
Livnium is a provably locally-contracting pseudo-gradient flow.
See runs/livnium_collapse_equation.md for the full derivation and empirical direction mismatch analysis (135.2° ± 2.5° between Euclidean and cosine gradients).
Citation
If you use this work in your research, please cite:
bibtex
1@misc{patil2026livnium,
2 author = {Patil, Chetan},
3 title = {Livnium: Energy-Guided Attractor Network (EGAN) for Natural Language Inference},
4 year = {2026},
5 publisher = {GitHub},
6 howpublished = {\url{https://github.com/chetanxpatil/livnium}},
7 note = {Model available at \url{https://huggingface.co/chetanxpatil/livnium-snli}}
8}
