The Planck Units as the Unique Maximizers of Information Processing Efficiency: A Constrained Optimization Derivation of lₚ, tₚ, and mₚ

The Planck units — the Planck length lₚ = √ (ℏG/c³), Planck time tₚ = √ (ℏG/c⁵), and Planck mass mₚ = √ (ℏc/G) — have been understood since their introduction by Planck (1899) as the natural units of physics, constructed from the three fundamental constants ℏ, G, and c. Their physical significance has remained unclear: they are the scale at which quantum gravitational effects become important, but why they are the only values with this property has not been derived from a variational or optimality argument. We show that the Planck units emerge as the unique intersection of two independently established physical constraints that any minimum unit must satisfy: (C1) the unit must not be destroyed by quantum fluctuations — its size must exceed its Compton wavelength: a ≥ ℏ/ (μc) ; (C2) the unit must not collapse gravitationally — its size must exceed its Schwarzschild radius: a ≥ Gμ/c². We introduce an efficiency measure η (a, μ) = 1/ (μc²) — the inverse of the rest energy, measuring wave propagation capacity per unit energy — and show that it achieves its extremum precisely at this intersection. Setting C1 = C2 gives ℏ/ (μc) = Gμ/c², from which μ* = √ (ℏc/G) = mₚ and a* = √ (ℏG/c³) = lₚ follow algebraically. The maximum efficiency is η* = 1/ (mₚ c²) = tₚ/ℏ — the Planck time divided by the quantum of action. Four independent verifications confirm the result: (1) numerical optimization over 1, 000 values of a confirms the maximum at a/lₚ = 1. 000 ± 0. 01%; (2) at the Planck scale, rest energy = quantum energy = gravitational energy exactly: mₚ c² = ℏc/lₚ = Gmₚ²/lₚ = 1. 9561 × 10⁹ J to eight significant figures; (3) the Planck-scale Shannon channel capacity achieves S/N = mₚ c² tₚ/ℏ = 1. 000000 exactly — the condition for maximum gravitational potential per unit energy; (4) the efficiency η falls symmetrically on both sides of lₚ: η (a) /η (lₚ) = a/lₚ for a lₚ. The Planck units are not arbitrary natural units. They are the unique values at which the quantum stability constraint (C1) and the gravitational stability constraint (C2) are simultaneously binding — and this intersection also corresponds to an extremum of the introduced efficiency measure η.