The Tetrahedron Protocol: A Geometric Framework for Unified Systems Theory Connecting SIC-POVM Quantum Measurement, Structural Rigidity, and Biological Coherence

This paper proposes that the regular tetrahedron serves as a universal design principle connecting four domains: quantum information theory, structural engineering, quantum biology, and network topology. The framework demonstrates that the geometric properties of the tetrahedron — specifically its equiangularity, isostatic rigidity, and informational completeness — manifest as a recurring structural signature across scales from subatomic measurement to macroscopic network architecture.

In quantum information theory, Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) arrange four measurement vectors as a regular tetrahedron inscribed in the Bloch sphere, satisfying the equiangular overlap condition |⟨ψⱼ|ψₖ⟩|² = 1/(d+1) = 1/3. This geometry enables full quantum state tomography from a single measurement basis, eliminating the sifting inefficiency and reference frame dependence of orthogonal protocols such as BB84.

In structural engineering, Maxwell's rigidity criterion (E ≥ 3V − 6) identifies the tetrahedron (V=4, E=6) as the minimum isostatically rigid three-dimensional structure — the simplest geometry that encloses volume and resists deformation without internal bracing.

In quantum biology, Fisher's Posner molecule hypothesis identifies calcium phosphate clusters (Ca₉(PO₄)₆) as potential carriers of quantum coherence in neural systems. The phosphate group PO₄ is itself a regular tetrahedron, with the tetrahedral bond angle satisfying cos(109.47°) = −1/3 — the same constant governing SIC-POVM overlap.

In network topology, Ollivier-Ricci curvature analysis distinguishes fragile hub-and-spoke architectures (negative curvature) from resilient mesh topologies (positive curvature), where triangulated connectivity provides redundant pathways resistant to single-node failure.

This framework is presented as a defensive publication establishing prior art in the public domain. The author is an independent researcher with 16 years of experience in submarine electrical engineering, specializing in three-phase power distribution and motor system topology. Applications to assistive technology design for neurodivergent communication systems are discussed.