The Math, Game Theory, and Physics Behind the Blockchain

We present a unified mathematical framework: the Information Nexus. Connecting distributed ledger security, stochastic volatility, and cosmological evolution through a single scalar information field S (x, t). The master inequality d (a³ρI) /dt ≥ 0, combined with an efficiency function η = e^-X and a universal critical threshold Xc = ln 2, generates three independent domains of application: (I) the exact security of Nakamoto consensus via the negative binomial correction; (II) a thermodynamic blockchain engine with three entropy channels and a functional lifespan equation; (III) the Inverse Correlation of Entropy and Efficiency with five attack channels; (IV) Thermo-Geometric Volatility with β = 3/2 forced by the cubic degree of the Weierstrass elliptic curve; (V) the Relativity of Information with dark matter as thermodynamic residue; (VI) the Scale-Invariant Nodular Model of Cosmology (SINMoC) with a seven-phase lifecycle; (VII) the Satoshi Singularity as a phase transition from computational to gravitational logic; (VIII) the ECC Spine connecting secp256k1 to the QCD-epoch elliptic curve; (IX) the Perry Correspondence — a bijective mapping across cosmic, ledger, and market domains. Twenty-seven testable predictions are derived, including Ω_Λ = ln 2 (currently 1. 15σ from observation) and a Landauer mass at confinement threshold dc = 17. 7 nm.