Special Relativity from the Tree of Continua: A Complete Derivation without Metrics, from Three Primitives through the Exponential Function

Special relativity is derived entirely from the Tree of Continua C and the exponen-tial function. No postulate of special relativity is assumed. No metric is introducedas a primitive at any point.The derivation rests on three objects, each traceable to the primitive of distin-guishability:IPG1. The exponential eψ ∈ C, defined by SN (ψ) −−→ eψ where SN = Nn=0 ψ /n!is finite rational arithmetic. The hyperbolic identity cosh2 ψ − sinh2 ψ = 1follows by four lines of arithmetic on eψ .2. The LDI inner product ⟨u, v⟩ = −αβ+hμν |Vphys uμ⊥ v⊥μν is the finite difference Hessian of the LDI scalar field Φn on the grid Nn−1 of integer jointperiods. The Lorentzian signature (−, +, +, +) is canonical (up to rescalingof the temporal unit) from the gauge direction of Φn . The spatial dimensionn − 1 = 3 is derived from the antichain structure of cylinder sets.3. The Lorentz transformation Λ(ψ), the unique linear isometry of the LDIinner product connected to the identity. Its existence and uniqueness are forcedby the hyperbolic identity: the isometry condition a2 − c2 = 1 is the hyperbolicidentity.All SR kinematics follow: invariant interval, time dilation, length contraction,velocity addition, mass-shell condition, Doppler factor, twin paradox, Poincaré group.Einstein’s two postulates are theorems. The Poincaré ball is the velocity space. Theten conserved quantities follow from Noether’s theorem on the grid — no integrals,no action principle.The foundation is a discrete grid. The continuous structures are IPG limits ofgrid computations. No metric appears as a foundation anywhere in the chain.