Primes and Circle Duality J; Arithmetic Rigidity of Covering Entropy– Prime Bounds, Higher-Dimensional Covering, and Quantum Phase Classification

Within the framework of information geometry and computable analysis, this
paper establishes three arithmetic applications of covering entropy theory. First,
using the composite expansion law, we derive an explicit information-theoretic
lower bound for the prime-counting function, establishing a coding correspondence
between the cardinality of the minimal incompressible covering skeleton and the
covering entropy. Second, we connect the ring of integers of a degree-r algebraic
number field via the Minkowski embedding to the r-dimensional torus, and within
the classical framework of Landau’s prime ideal theorem and Chebotarev’s density
theorem, we establish a cross-domain information-coding correspondence between
the prime ideal theorem and the asymptotic density of arithmetic covering skele
tons on higher-dimensional tori. Third, using the rigidity gap of fusion categories,
we rigorously compute the lower bound of the information distortion from the Fi
bonacci modular tensor category to any compact group representation category
as 1, thereby demonstrating that the Fibonacci topological phase cannot be sim
ulated by any classical symmetry phase with information distortion less than 1.
This paper explicitly declares the two-axiom core of the underlying self-consistent
logic—the axiom of information conservation and the axiom of computability—and
introduces a conditional depth-grading framework that clearly distinguishes encod
ing readability from logical equivalence. We explicitly state that we do not prove
the Riemann hypothesis, and all statements involving that hypothesis are marked
as open problems.