The ln2 Entity and the Dimensional Power Law: From Biology to Quantum

The unification of continuum and quantum descriptions remains one of the deepest unresolved problems in theoretical physics. Here we demonstrate that a scalar field equation, established in a companion preprint, is governed axiomatically by the transcendental constant ln 2. We elucidate the ontological status of ln2 and report a rigorous power-law regularity: the exponent of ln2 in a boundary function strictly dictates the spatial dimension of the physical regime it encloses. The thresholds 1 − ln2 ≈ 0.307 and Rc = (2 ln2 − 1)/(ln 2)² ≈ 0.804 emerge as exact algebraic consequences of the field equation, demarcating three-dimensional Euclidean space, four-dimensional relativistic spacetime, and five-dimensional (or higher) quantum phase space. Cross-system validation spanning open biological systems, superconducting qubits, and metropolitan-scale quantum networks confirms that physical systems converge to these predicted limits without empirical adjustment. We conclude that ln2 is the universal information–energy conversion unit and the generative constant of dimensional hierarchy.