FERNEK-Ω: Geometric Invariants and Epistemic Boundaries in Latent Space — Complete Corpus V1–V-Final

Complete corpus of the FERNEK-Ω protocol across 28 experimental sessions (V1: Sessions 1–14, V2: Sessions 1–5, V3: Sessions 1–4, V4: Session 1, V-Final). 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: drift 29.4× lower than baseline; adversarial separation 4.19×; dual-head collapse rate 0.8% vs 8.8%. 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, motivated by Balestriero & LeCun (LeJEPA, arXiv:2511.08544). S²³ trained embeddings are significantly more isotropic than the 384D baseline (t=42.878, p<0.0001), but SIGReg regularisation achieves higher isotropy (0.512 vs 0.157, effective rank 23.96/24). Sessions 4–5 establish the central epistemic finding: isotropy, effective rank, and opposition angle are all invariant to semantic validity and logical coherence (p>0.40 for all discrimination tests).

V3 introduces geometric tension τ (Sessions 1–3) and its temporal derivative Δτ (Session 4). τ discriminates structured semantic content from random nonsense (p<0.01, replicated across three sessions) but does not discriminate complex truths from coherent lies (p=0.948, d=0.008 — structural isomorphism confirmed). Δτ = ||z(T+1) − z(T)|| detects temporal contradiction with the largest effect size in the corpus (d=2.419, p<10⁻⁴): an agent that contradicts itself produces a measurable geometric rupture. FERNEK-Ω is a kinematic coherence monitor, not a lie detector.

V4 tests whether internal logical contradiction manifests as instability under Monte Carlo Dropout (N=50 stochastic passes). Result: coherent lies and complex truths are geometrically isomorphic at the stochastic level (p=0.975, d=0.008). The epistemic boundary is confirmed at three levels: static (p=0.948), stochastic (p=0.975), and temporal (Δτ detects contradiction across time only).

V-Final tests cross-domain universality of Δτ. An encoder trained on agent/system sentences is tested on three unseen domains (Biology, Law, Physics). Effect sizes attenuate significantly (d=0.37–0.71 vs d=2.419 in training domain). Conclusion: Δτ is a domain-calibrated coherence monitor, not a universal geometric law. Cross-domain deployment requires domain-specific encoder calibration.

Central finding across all versions: Geometry is a neutral channel. It transports structured content — true or false — with equal fidelity. It detects incoherence across time within a calibrated domain. It cannot detect falsity at any single instant, at any level of analysis. FERNEK-Ω is a metrological instrument: it measures the geometry of reasoning with precision. It does not judge.

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

Keywords: S²³, Leech lattice, spectral isotropy, SIGReg, geometric tension τ, temporal consistency Δτ, MC-Dropout, epistemic boundary, kinematic coherence monitor, domain calibration, FERNEK