Topological Coherence

Toroidal attention constraints for LLM coherence — in-sample training-dynamics effects; inference-time hallucination reduction is a null result (see below)

Sylvain Cormier | Paraxiom Research | 2026

Status (v7 — March 2026 Replication Update)

Inference-time logit bias: A comprehensive 6-phase replication found the v2 TruthfulQA improvements (+0.2pp to +2.8pp) were within LLM judge sampling variance. Exact T&I replication: −2.0pp, p=0.22 (not significant).

What works:

• Prompt hardening: −14pp hallucination reduction (p=0.05) — the active ingredient in Coherence Shield
• Training-time topology: 28x lower drift than random sparsity in toy model (topology matters, not just sparsity)
• Library primitives: Tonnetz geometry, attention masks, spectral gap, drift measurement all valid

Recommended direction: Training-time Karmonic spectral regularization (untested, promising based on toy model)

• Paper: DOI: 10.5281/zenodo.18516477

Installation

pip install topological-coherence

# With HuggingFace transformers support
pip install topological-coherence[hf]

Quick Start: Toroidal Logit Bias

Drop-in logit processor for any HuggingFace model — no fine-tuning required:

from topological_coherence import ToroidalLogitProcessor
from transformers import AutoModelForCausalLM, AutoTokenizer

model = AutoModelForCausalLM.from_pretrained("gpt2")
tokenizer = AutoTokenizer.from_pretrained("gpt2")

processor = ToroidalLogitProcessor(grid_size=12, radius=2.0, alpha=0.3)

inputs = tokenizer("The quantum nature of", return_tensors="pt")
outputs = model.generate(
    **inputs,
    logits_processor=[processor],
    max_new_tokens=100
)
print(tokenizer.decode(outputs[0]))

Core API

Tonnetz Geometry

from topological_coherence import Tonnetz, distance_matrix

# Create a 12x12 torus topology
t = Tonnetz(grid_size=12)
t.distance(0, 5)        # L1 toroidal distance with wraparound
t.spectral_gap()         # First eigenvalue of the torus Laplacian

# Vectorized distance matrix (numpy, fast)
dm = distance_matrix(n_tokens=64, grid_size=12)  # (64, 64)

Attention Masks (3 variants)

from topological_coherence import ToroidalMask, sinkhorn_knopp

mask = ToroidalMask.hybrid(seq_len=64, radius=2.0, alpha=1.0)   # default
mask = ToroidalMask.hard_cutoff(seq_len=64, radius=2.0)          # binary
mask = ToroidalMask.soft_exponential(seq_len=64, alpha=1.0)      # smooth decay

tensor = mask.to_tensor()                     # torch.Tensor for attention
ds = sinkhorn_knopp(tensor, n_iters=50)       # project to doubly-stochastic

Drift Measurement

from topological_coherence import DriftMeter

meter = DriftMeter(threshold=2, grid_size=12)
meter.record(pred=5, target=8)
meter.record(pred=5, target=100)
print(f"Drift rate: {meter.rate():.3f}")

Toroidal Attention (PyTorch)

from topological_coherence import ToroidalAttention, TinyTransformer

# Drop-in attention replacement
attn = ToroidalAttention(d_model=64, n_heads=4, max_seq_len=64)

# Full demo transformer with swappable attention
model = TinyTransformer(
    vocab_size=144, d_model=64, n_heads=4,
    attention_type="toroidal"  # or "baseline", "random"
)

Theory

Hallucination is a geometry problem. Unconstrained latent dynamics permit arbitrary drift through embedding space. Toroidal constraints provide a constant spectral gap that suppresses non-resonant modes:

λ₁ = 2 - 2cos(2π/N) = Θ(1)    for fixed grid size N

This bounds semantic drift without reducing model capacity.

Hierarchy: mHC (Birkhoff) ⊂ ERLHS (Hamiltonian) ⊂ Karmonic (Toroidal + Spectral)

Links

• Paper (Zenodo)
• Toroidal Logit Bias Paper
• Live Demo (HuggingFace)
• Source (GitHub)
• Rust Crate (crates.io)

Citation

@misc{cormier2026topological,
  author = {Cormier, Sylvain},
  title = {Topological Constraints for Coherent Language Models},
  year = {2026},
  publisher = {Zenodo},
  doi = {10.5281/zenodo.18187835}
}

License

Apache-2.0