Reflexive Information Geometry: A Quasicrystalline Substrate for Physical Law

We present the Reflexive Information Geometry (RIG) framework, in which physical law emerges from a five-dimensional quasicrystalline substrate (Z⁵) whose geometry is governed by the golden ratio φ = (1+√5) /2 as a self-referential substitution constant. The substrate is not postulated as an ansatz but is the unique structure consistent with the self-referential equation x = 1 + 1/x, making φ a fixed point of the geometry rather than a free parameter. We derive: (i) the Z⁵ frame identity Σ eₖ ⊗ eₖ = (5/2) I exactly; (ii) the minimum stable vortex spans two adjacent Z⁵ sectors, with binding energy 1. 5% below the threshold; (iii) the eigenvalue ratio λ₂/λ₁ = φ² exactly from the Z⁵ Laplacian; (iv) the factor √2 in the Brannen lepton mass parameterization from the Z⁵ coupling geometry; and (v) the Koide formula for charged lepton masses as a consequence of three free vortex phase orientations in the Z⁵ frame, separated by 2π/3. The framework predicts the Bohr radius as log_φ (1/α) ≈ 10. 2 substrate levels. An independent empirical test of the φ-core dark halo profile v (r) = v₀ r/√ (φ²r₀² + r²) against 172 SPARC galaxies yields median χ² = 1. 221, outperforming the pseudo-isothermal sphere (median 1. 293) on 108/172 galaxies with φ fixed and not fitted (Berry 2026). We state explicitly which results are exact derivations, which are motivated proposals requiring further proof, and which are open problems. We invite collaboration on the identified open derivations, in particular: the substrate equation of motion, the Brannen angle from Z⁵ geometry, and the renormalization group flow of the effective coupling.