Basic Research on Number-Theoretic Integration of Physical Constants

This paper defines the foundations of BNC Theory, which integrally describes the origins of the fine-structure constant alpha, the cosmological constant Lambda, and the gravitational constant G—the greatest unsolved problems in modern physics—from the perspective of geometric information theory. We define physical constants not as independent entities, but as cross-sections of information structures that have crystallized into 4-dimensional spacetime from a 6-dimensional possibility manifold Omega_6D via Phase Alignment. In this paper, we theoretically construct the following three points in particular: Rigorous Definition of the Torque Wall: We formulate the physical cost of determining information as an entropy limit within an operator algebraic framework and couple it with the non-locality of the K-term. Topological Definition of Semantic Closure: We redefine the self-consistency of the universe as a vanishing condition of topological invariants based on the Atiyah-Singer index theorem, eliminating circular reasoning. Establishment of the Standard Form of the K-Term: We uniquely define the non-local interaction term K[u] as a geometric correlation kernel within an operator algebraic framework and rigorously separate its structure into a Local beta-component and a Non-local alpha-component. This theory suggests that the universe is a self-referential computational system that optimizes physical constants to achieve a self-consistent Semantic Closure. It positions the phenomenon of Prime Number Crystallization, which occurs when an observer is highly phase-aligned with spacetime, as a new frontier in physics.