H2E Sheriff: Mathematical Derivation of Universal Safety Constants Including the Lambda Spectral Complementarity Theorem and Applications

This paper, authored by Frank Morales Aguilera in May 2026, presents a deterministic mathematical framework for AI governance known as the H2E (Human-to-Expert) Sheriff. The core of this system is the derivation of universal safety constants—specifically the Safety Threshold ($\Lambda$) and the Fixed-Point Exponent ($\alpha^*$)—derived solely from the first six prime numbers $\{2, 3, 5, 7, 11, 13\}$. By basing these constants on primes rather than empirical tuning or hardcoding, the author proposes a "zero-error capacity" safety layer for multimodal AI systems.

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Core Mathematical Framework

The derivation is built upon two primary theorems:

• Lambda Spectral Complementarity Theorem: This theorem establishes the Safety Threshold ($\Lambda \approx 0.9785$). It utilizes the Euler attenuation product ($I$), which measures spectral energy loss across prime channels. The theorem posits a conservation law where the energy lost ($I$) plus the energy retained ($\Lambda$) must equal 1.

• H2E Fixed-Point Theorem: This proves the existence of a unique exponent ($\alpha^* \approx 1.0001$) where the spectral norm of a prime-circulant matrix ($L_{13}$) exactly matches the Safety Threshold. The value is found using an algorithmically derived binary search that requires no human intervention or domain knowledge.

Real-World Applications

The paper demonstrates the versatility of these constants by applying them to three distinct domains, using the same prime-derived thresholds for each:

1. Sport Simulation (FIFA World Cup 2026): Governs 104 matches by ensuring the AI's tactical execution aligns with established "DNA ground truth" vectors.

2. Financial Signal Governance: Uses a "Spectral Market Sieve" to filter market noise. Decisions to permit market entry are only validated if the signal purity (SROI) exceeds the Safety Threshold.

3. Atmospheric Hazard Detection: Applies the framework to hurricane detection. In the provided test case, the system triggered a "hard stop" because the storm signature was too buried in sensor turbulence to meet the safety requirement.

Determinism and Reproducibility

A significant portion of the paper emphasizes that these results are fully reproducible. By using a fixed seed (123) and deriving all values from the first six primes, the author ensures that identical constants are produced across any hardware or environment. The H2E Sheriff thus operates as a rigid governance layer where every AI decision is either mathematically certified as safe or immediately rejected.