The Toroidal Geometry of LLM Representations: From Detection to Karmonic-Guided Alignment

We detect toroidal topology in LLM hidden states via persistent homology and
  introduce Karmonic spectral filtering as a training-time regularizer that
  outperforms Sinkhorn optimal transport for truthfulness alignment.

  Torus Detection (Qwen 2.5-0.5B, TruthfulQA):

  Layer 0 (embedding): β₁=32, β₂=8, Angular Sep=120.7°
  Layer 12 (middle): β₁=37, β₂=21, Angular Sep=35.8°
  Layer 23 (final): β₁=25, β₂=4, Angular Sep=35.0°

  Alignment Results (DPO + LoRA, 200 steps, TruthfulQA 817 samples):

  Untrained baseline: MC1=0.278, MC2=0.407, PPL=18.06
  DPO only: MC1=0.486, MC2=0.554, PPL=18.24
  DPO + SAMI + Sinkhorn OT: MC1=0.448, MC2=0.499, PPL=18.18
  DPO + SAMI + Karmonic: MC1=0.461, MC2=0.544, PPL=18.22
  DPO + Karmonic: MC1=0.468, MC2=0.531, PPL=18.21

  Key findings:
  - Karmonic > Sinkhorn OT by +1.3pp MC1 and +4.4pp MC2 in matched conditions
  - DPO + Karmonic achieves MC1=0.468 (+19pp over untrained baseline)
  - Toroidal topology detected across all layers (β₁=23-37)
  - Perplexity stable (18.18-18.24) — no quality degradation

  Scope: Empirical. Detects toroidal topology and shows torus-aware
  regularization outperforms geometry-agnostic drift control. No ontological
  claims.

  Code and data: https://github.com/Paraxiom/topological-coherence