FERNEK-Ω: Geometric Invariants and Epistemic Boundaries in Latent Space

Complete corpus of the FERNEK-Ω protocol across 19 experimental sessions (V1: Sessions 1–14, V2: Sessions 1–5). All experiments run on Google Colab CPU, no GPU required.

V1 establishes two absolute geometric invariants of the 24-dimensional sphere S²³: topological rank r=24 and saturation exponent β≈−0.131±0.014, reproduced across 72 independent runs (Δ=0.0019). Key results: inter-agent drift 29.4× lower than unconstrained baseline; adversarial separation 4.19× higher (Shapira et al.); dual-head collapse rate 0.8% vs 8.8% baseline. Negative results (Sessions 11–14) define the decisional boundary: S²³ stabilises geometric position but does not protect decision boundaries under Gaussian perturbation, does not improve vector quantisation, and acts as a lossless pass-through (Geometric Sandwich, p=0.751).

V2 introduces spectral isotropy as a second axis, motivated by Balestriero & LeCun (LeJEPA, arXiv:2511.08544), which proves the isotropic Gaussian is the unique optimal latent target. S²³ trained embeddings are significantly more isotropic than the 384D baseline (t=42.878, p<0.0001), but explicit SIGReg regularisation achieves higher isotropy (0.512 vs 0.157) with near-complete dimensional coverage (effective rank 23.96/24). A hybrid S²³+SIGReg partially bridges the gap. Sessions 4–5 establish the central epistemological finding: geometric structure — isotropy, effective rank, or opposition angle — is invariant to semantic validity and logical coherence (p>0.40 for all discrimination tests).

Central finding: Geometry is a channel, not a truth filter. FERNEK-Ω is a metrological instrument for latent geometry: it measures with precision; it does not judge. This result directly addresses the Geometric Morality Fallacy implicit in current AI alignment research.

References: LeJEPA (arXiv:2511.08544), Goldfeder & LeCun SAI (arXiv:2602.23643), Roig RIKER (arXiv:2603.08274), Song et al. reasoning failures (arXiv:2602.06176), Lying to Win (arXiv:2603.07202).

Keywords: S²³, Leech lattice, spectral isotropy, SIGReg, geometric invariants, latent space, epistemic boundary, SAI, FERNEK