Paper Zero: The L-System: A Single-Dimension Foundation for the Structural Isomorphism of Physics

The SI unit system encodes the historical separation of mechanics, thermodynamics, and electromagnetism into dimensionally independent quantities. We show that this separation obscures a deeper structural unity. Starting from a single postulate — the thermo-electric equivalence 1 J ≡ 1 C × 1 K — we derive a system in which all physical dimensions reduce to powers of a single fundamental dimension: Length. In this "L-system" the Boltzmann constant acquires the dimension of area ([kB] = L²), temperature becomes inverse volume ([K] = L⁻³), and charge becomes area ([C] = L²). We verify dimensional consistency against 16 equations spanning thermodynamics, electromagnetism, mechanics, general relativity, cosmology, and quantum field theory. The L-system admits two dual representations: Holographic (Boundary) Representation: Each sector's charge is measured on the gravitational Planck area ℓP², yielding [G] = [kB] = [e] = L² and [ε₀] = L⁴, [εg] = L⁻². Isomorphic (Bulk) Representation: Each sector is measured on its own natural surface area AX = 4π(ΦX/c)², whereupon all charges reduce to L⁻¹, all vacuum permittivities reduce to L⁻², and all potentials (c², V, T) reduce to L⁰ — pure dimensionless geometric gradients. The passage between the two representations is a holographic projection. The L⁶ hierarchy between εg and ε₀ in the boundary representation vanishes entirely in the bulk representation, resolving the Hierarchy Problem as a gauge artefact of the measurement background. We further show that the Fermi constant GF reduces to the electroweak area ℓEW²/√2, and the QCD string tension σ reduces to L⁻² (inverse area), identifying confinement as a flux tube whose fixed cross-section is the geometric dual of the L² charges. A complete dimensional map of all four interaction sectors emerges, with the power spectrum L⁻⁴ to L⁺⁴ reflecting the four-dimensionality of spacetime. Keywords: dimensional analysis, natural units, Planck units, structural isomorphism, holographic duality, hierarchy problem, gauge theory Related papers: This paper provides the dimensional foundation for Papers I–VI of the QGD programme.