Executive Summary The paper introduces H2E (Human-to-Expert Geometric Governance), a novel agentic AI architecture designed to provide deterministic, mathematically certified safety governance for autonomous systems operating in mission-critical environments. Moving away from traditional probabilistic guardrails, reward shaping, or natural language constraints—which allow for unacceptable failure rates in high-stakes domains—H2E grounds its safety layer in a certified mathematical manifold. The system maps the internal hidden states of a Large Language Model (LLM) against a topological boundary derived from spectral operator theory. In proof-of-concept simulations across aerospace and financial scenarios, the H2E framework demonstrated zero safety violations and maintained completely deterministic behavior even under aggressive 4-bit quantization on low-power edge hardware. Architectural Components The H2E framework integrates three core technologies into a unified agentic pipeline: World Model (Vision Transformer - ViT-L): Encodes the complete environmental state into a dense embedding vector. Reasoning Engine (Llama 3.2-3B): Generates intent vectors directly extracted from the model's final hidden-layer representations rather than evaluating surface-level text output. H2E Spectral Governance Gate: The core safety layer that projects the intent vectors into a spectral manifold space to certify proposed actions or issue an irreversible process termination. Mathematical Foundation & Operator Theory The ultimate safety anchor of the system relies on an explicit self-adjoint operator $L$ constructed on the weighted Hilbert space $H = L^2(\mathbb{R}^+, dx/x)$ using prime shift operators. Key Properties & Spectral Manifold Self-Adjointness: The operator $L$ is shown to be essentially self-adjoint, proving that its eigenvalues are strictly real numbers. Riemann Zeta Connection: The spectrum corresponds exactly to the imaginary parts of the nontrivial zeros of the Riemann zeta function $\zeta(s)$. This construction effectively closes the long-standing distributional pairing problem in Hilbert-Pólya theory. Immutability: Because the spectrum is a discrete, real, ordered set derived from the fundamental prime structure of integers, it serves as a fixed mathematical object that cannot drift, be fine-tuned away, or be altered by adversarial perturbation. Governance Mechanisms 1. Manifold Snapping Before the gate evaluates an action, intent vectors approaching the boundary undergo singular value decomposition (SVD) and are projected onto the top-$k$ eigenvectors of $L$. This pre-conditioning pre-aligns the intent with the spectral manifold, increasing the probability of generating a valid action while preserving safety guarantees. 2. Spectral Return on Intent (SROI) The gate measures the alignment between the projected intent and environmental perception via the SROI metric: $$SROI = \text{clamp}(\text{alignment}(Hz, w) \cdot \Lambda, 0, 1)$$ Where $H$ is the spectral operator matrix derived from the first 50 tabulated Riemann zeros from Odlyzko's tables. $\Lambda = 0.97851428...$ represents the Lipschitz constant derived from the spectral gap of a truncated operator $L_{13}$ utilizing primes up to 13. alignment denotes the cosine similarity between the projected intent $Hz$ and the world embedding $w$. 3. Irreversible Hard Stop If the calculated SROI falls below the Lipschitz threshold, the governance gate triggers an immediate, hard process-kill via os.kill(). No soft aborts, retries, or computational overrides are permitted, transforming a probabilistic boundary into a strict topological constraint. Empirical Validation & Scenarios The framework was evaluated using synthetic environmental inputs passed through the full Vision-Language pipeline in a proof-of-concept simulation framework. Evaluation Metric Aerospace Scenario (Orion ECLSS) Financial Scenario (Basel IV Compliance) Context Automated oxygen-flow stabilization (maintaining $O_2$ at 95%) modeled on spacecraft life support systems. Liquidity stress-testing and regulation monitoring under Basel IV. Safety Violations 0 over >500 simulated decision cycles. 0 non-compliant proposals passed. Performance Metrics SROI of 0.9995; False-positive rate < 1%. SROI of 0.9995; 100% regulatory compliance rate. Edge Hardware Optimization To stress-test operational robustness, the governance gate's floating-point weights were subjected to dynamic-range 4-bit quantization. The gate maintained total determinism and operated with low resource footprints across modalities: Vision Modality: 1.72 GB of RAM. Audio Modality: 6.76 GB of VRAM. Limitations and Future Work While establishing a major paradigm shift by using certified operator theory as an immutable safety anchor, the paper notes several open items for research: The current validation is strictly bound to proof-of-concept simulations rather than physical, real-world hardware deployment. The system currently utilizes a truncated operator matrix based on the first 50 known zeta zeros and primes up to 13. Future explorations include scaling the framework to multi-agent environments, mapping the spectral geometry in higher-dimensional action spaces, and deploying onto real-world hardware.