Gravitational and Planck Constants as Mathematical Necessities: First Derivation of G and ℏ from Geometric, Topological, and Information-Theoretic Structure

We present the first complete explanation of why the gravitational constant G and the Planck constant ℏ must have the values they do. Instead of treating them as arbitrary inputs to physics, we show that they arise naturally from the geometry, topology, and information structure of spacetime itself. Our argument uses three independent lines of reasoning.

First, basic geometric facts — the 4π structure of spheres and the 2π structure of closed loops — together with the crossover between gravitational and quantum behaviour, fix the fundamental Planck scales without requiring any adjustable parameters. Second, limits on how much information can fit into a region of space show that there is a smallest meaningful length scale where the usual continuum breaks down, and this scale matches the Planck length. Third, a thermodynamic view of spacetime, using heat flow and entropy near local horizons, reproduces Einstein’s equations and again singles out the same Planck scale.

Because all three approaches converge on the same answer, we argue that G and ℏ are not empirical accidents, but follow inevitably from the underlying structure of spacetime. Our framework connects naturally to existing ideas such as loop quantum gravity, holography, and asymptotic safety, while giving clear predictions for where new physics should appear. In short, this work offers the first unified explanation for the origin of the most fundamental constants in nature.