The Geometrical Derivation of Born’s Rule: From Reciprocal Closure and Compact Return to Conjugate Chords and Quadratic Outcome Weights

The Born rule assigns normalized outcome weights by Pj = = ||Πjψ||². This paper derives that finite-dimensional weight law within an explicit geometrical closure architecture, without assuming a complex-amplitude representation or a positive-definite quadratic norm at the entrance to the derivation. The construction begins from a reciprocal logarithmic pair r±(θ) = R e±κθ, with r+r− = R². A compact-return quadratic-unity theorem is then proved. Any non-trivial continuous one-parameter linear action on a real two-dimensional carrier with finite return, T(τ) = I, generates an invariant positive-definite bilinear form by averaging over one return cycle. The form is unique up to positive scale. In its orthonormal coordinates the action is SO(2), its normalized generator satisfies J² = −I, and the carrier acquires a complex structure. The reciprocal product r+r− = R² fixes the remaining scale of the derived quadratic unity. A continuous multiplicative character of the reciprocal branch ratio ρ = r+/r− into the derived U(1) phase circle has the form χλ(ρ) = exp(−iλ ln ρ). Branch exchange ρ -> ρ−1 therefore becomes complex conjugation. Writing Ψ = R exp(−iα), Ψ* = R exp(+iα), α = λ ln ρ, the chord joining the two conjugate points decomposes the invariant radius into orthogonal components M = R cos α, H = R sin α, so that M² + H² = R². If δ = 2α is the central conjugate separation, non-negative unit-normalized additivity under orthogonal composition uniquely fixes the weight functional as W(a) = a², giving P0 = cos²(δ/2), P1 = sin²(δ/2), P0 + P1 = 1. Repeated minimal adjoining of distinguishable alternatives preserves the same binary quadratic closure law and generates Hn ≅ C^n, Qn(c1, ..., cn) = Σ |cj|², with the corresponding finite-dimensional Hermitian inner product recovered from Qn by polarization rather than separately assumed. Repeatability of a linear resolution map gives idempotence, P² = P. Requiring the derived quadratic closure to decompose between a resolved component and its complement forces P = P†; orthogonal projectors are therefore derived rather than separately postulated. For a complete finite family, completeness then forces pairwise orthogonality. For normalized ψ, Pj = ||Πjψ||² = , and Σj Pj = 1. The derivation is complete at the level of the finite-dimensional normalized Born weight law under the stated structural premises. It does not establish which mathematical representation is physically constitutive of a quantum state, select a concrete measurement basis, explain individual outcome occurrence, or derive terminal event frequencies without an additional realization dynamics.