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

⚠️ CORRECTION NOTICE (July 2026): key claims of this v1 are RETRACTED in version 2 (10.5281/zenodo.21227351). Preregistered multi-seed replications on two architectures returned null results for the Karmonic-vs-Sinkhorn-OT advantage, and the β₁ cycle-count statistic failed a matched-covariance Gaussian null. Bistable DPO training collapse explains the earlier single-run signals. Please read and cite v2, which documents what stands and what does not. 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