A Capacity-Based Basis for Planck Quantities and Gravitational Coupling

This paper presents an exact algebraic reformulation of the Planck system in which all standard Planck quantities and the gravitational coupling reduce to a unique, closed, and minimal integer-exponent monomial basis.By factorising the standard Planck definitions through the Planck force and eliminating redundant generators, all quantities are shown to take the universal formQ = tₚᵃ Aₚᵇ cⁿ , with (a, b, n) ∈ ℤ,where the capacity constant Aₚ ≡ ħc emerges necessarily from the algebraic structure rather than being introduced by postulate.Within this basis, the gravitational coupling satisfies the exact identityG = Aₚ / mₚ²,demonstrating that it is fully determined by capacity and mass realisation rather than acting as an independent fundamental constant. Scale enters physics only through counting relative to this invariant Planck-scale structure.The result constitutes a closed algebraic proof of minimality and necessity within the Planck system, providing a rigorous bridge between exact Planck-scale invariants and macroscopic physics without modifying or extending existing physical laws.