This paper establishes a rigorous isomorphism between probability mathematics and informational resistance. The fundamental translation Ω(P) = −ln(P) maps probability to resistance, transforming multiplication into addition. This correspondence extends across four equivalent representations (natural language, probability theory, circuit topology, and prime coordinate vectors) which function as a Rosetta Stone for discrete mathematics: the same computation can be expressed in any layer and translated exactly to the others.Boolean logic emerges from circuit topology: AND as series (resistances add), OR as parallel (conductances add), and NOT as phase interference. We prove the framework's validity through two complete worked examples. First, we derive the probability that two random integers are coprime, obtaining the known result 6/π² by traversing all four representation layers. Second, we demonstrate that relational database operations (JOIN, UNION, INTERSECTION) map directly to number-theoretic operations (GCD, LCM). We establish Turing completeness by constructing an explicit simulation of counter machines using prime exponents as registers. The framework reveals that prime numbers are precisely the irreducible elements of this informational structure, configurations whose resistance cannot be decomposed into sums of smaller resistances. This characterization, combined with the Gödel-style encoding of data through prime coordinates, suggests that any sufficiently powerful computational system must rediscover the primes as a structural necessity.