Quantum Mechanics as Cone-Induced Curvature: Rigidity Theorems for Linearity and the Born Rule

We prove that quantum mechanics is the unique theory compatible with normalization- induced curvature on complex Hilbert space. Two rigidity theorems establish: (1) Lin- earity: continuous, reversible, scale-invariant flows necessarily have linear generators, and (2) Born rule: the assignment P (ϕ|ψ) = |⟨ϕ, ψ⟩|2 is the unique probability mea- sure compatible with cone geometry, unitary invariance, and orthogonal additivity. These are not interpretations but uniqueness results—no other mathematical struc- tures are possible given the geometric assumptions. We derive the Schr¨odinger equa- tion, interference, Berry phase, and uncertainty relations as consequences of the master equation CurvC(X, Y ) = [X, Y ] − ⟨∇N,[X,Y ]⟩ ⟨∇N,x⟩ x applied to the quantum cone. The re- construction inverts standard quantum foundations: rather than postulating Hilbert space structure, we show it emerges from reversible dynamics on normalized states.