The Mathematics of Self-Regenerating Cryptographic Primitives

Abstract

Traditional digital trust architectures suffer from the “Library Problem”: dependency on pre-compiled, static lookup tables or binaries that must be trusted blindly, creating supply-chain vulnerabilities. This paper proposes a paradigm shift to Intrinsic Trust, where encoding infrastructure is mathematically regenerated at runtime rather than distributed. We introduce the 0MXI Calculus, a deterministic lattice system anchored on universal transcendental constants:the golden ratio Φ ≈ 1.618033988749895 and π ≈ 3.141592653589793, with a contraction ratio λ ≈ 0.339949771344778. Operations on a quantized F15 lattice ensure cross-platform determinism, bounded by a Prime Boundary Horizon (N = 23) that guarantees injective reversibility (Theorems 1 and 2).This framework underpins TreeOS, an operating system that bootstraps from a “Math Root-of-Trust” via autogenesis, regenerating a bijective Tick Table for byte encoding without stored dependencies. TreeBABEL, the verifiable data transport protocol, packages data as JSON artifacts with mathematical manifests for independent receiver validation. Extending this, the VMEM Node Architecture transforms online repositories into externalized memory banks, enabling AI models to scrape and derive OS state on demand, eliminating internal weight bloat and static knowledge cutoffs.We demonstrate adaptability to constrained ledgers (e.g., 280-character limits) for efficient chunking. Through rigorous proofs and a Python reference implementation, we show that trust can be calculated, not stored, decoupling systems from physical hardware and fostering entropy-neutral, zero-trust computation.