The Planck Units as Physical Necessity A Six-Paper Series on Minimum Length, Time, and Mass from the Requirement of Physical Finiteness

This collection presents six papers establishing a two-directional argument for Planck-scale discreteness of spacetime. The observational foundation: no infinite physical quantity has ever been measured, and the universe's age is finite (T = 4.354 × 10¹⁷ s). Every divergent physical equation produces infinities under the continuous limit dx→0 — the assumption that lengths and times have no lower bound. These papers argue this assumption fails at the Planck scale.Papers 54, 55, and 56 invert 21 established physical equations from six domains, demanding finite outputs and solving for minimum time, mass, and length. Seventeen of 21 equations (81%) converge on their respective Planck units without those units being assumed as input. Three equations yield exactly mₚ; three yield exactly lₚ. Paper 57 synthesises the trilogy, showing that the 81% convergence rate maps precisely onto the structure of fundamental interactions — complete for quantum-gravitational domains (QM, GR, SR), zero for electromagnetism (yielding α × Planck unit, physically explained). The convergence is not coincidence but the signature of a statistical fixed point. Paper 2 derives the Heisenberg uncertainty principle from Planck-scale discretisation via the discrete Fourier transform, with numerical verification ratio 1.00000000. Paper 3 shows that the singularities of general relativity are artefacts of applying differential calculus (dx→0) to a discrete world: replacing dx→0 with Δx≥lₚ gives finite curvature K(lₚ) = 5.87×10²¹⁸ m⁻⁴ at every formerly singular point. The Planck units are not assumed; they are derived as outputs in Papers 54–57, and their existence implies major results of physics in Papers 2 and 3.