Zero-Shot Geometric Probing Reveals Universal Cognitive Manifolds in Large Language Models

In this publication, we present a simple, zero-shot 3D geometric probing method that elicits highly consistent semantic manifolds from three divergent large language models (Gemma-3-27B, Llama-3.3-70B, GPT-OSS-120B) using only a single user prompt on consumer hardware via Ollama (no system prompts, fine-tuning, or steering). The models independently assign 3D coordinates to concept sets across 11 domains, recovering canonical human cognitive structures with near-perfect topological convergence: perceptual hue wheel (colors), temporal arrow with duration pillar (time), valence-arousal circumplex (emotions), warmth-competence quadrants (social roles), sacred-profane polarity (moral concepts), and physical-abstract threat opposition. Mid-density domains exhibit stable perceptual/cultural gradients, while sparse/absurd concepts fragment at manifold boundaries. Larger models extend invariant magnitudes without topological deviation. Japanese emotion probes recover identical circumplex structure to English, confirming deep language-general universality. A reverse probe anchored at "artificial intelligence" reveals shared future/transcendence-positive vs. existential risk/ethics-negative geometry, with human contrast orthogonal. All raw materials are included for full reproducibility: complete prompt texts, raw model outputs (JSON/TXT), extracted coordinate tables, Matplotlib visualization scripts (Python), generated 3D scatter plots (PNG), Open WebUI model configurations, and hardware/setup notes. This work demonstrates that zero-shot geometric probing is a powerful, minimal-intervention tool for diagnosing pretrained semantic organization and its scaling properties in LLMs. Abstract We introduce a zero-shot 3D geometric probing method that elicits highly consistent semantic manifolds from divergent LLM architectures (Gemma-3-27B, Llama-3.3-70B, GPT-OSS-120B) without system prompts or fine-tuning. Models independently assign relative coordinates to concept sets, recovering canonical structures: perceptual hue wheel (colors), temporal arrow with duration pillar (time), valence-arousal circumplex (emotions), warmth-competence quadrants (social roles), sacred-profane polarity (moral concepts), and physical-abstract threat opposition. Mid-density domains (food, deception, wealth) show stable perceptual/cultural gradients; sparse absurd concepts fragment at manifold boundaries. Convergence is near-perfect on perceptual/psychological cores. Larger models extend invariant magnitudes without topological deviation, amplifying nuance in variable domains. Japanese emotion probes recover identical circumplex structure to English, confirming language-general universality. Reverse probe from “artificial intelligence” reveals shared future-positive/risk-negative geometry. Runs on consumer hardware via Ollama. Zero-shot geometric probing emerges as a precise, reproducible tool for diagnosing pretrained semantic structure in LLMs.