Universal Structural Potentiality: A Rigorous Categorical and Functional-Analytic Framework for the Unification of Physical Theories, With the Schrodinger Derivation Theorem ¨ and the Arithmetic Potentiality Conjecture Francis Henry Ortiz Hidalgo

We develop the Universal Structural Potentiality (USP) framework as a mathematically rigorous programme for the unification of physical theories. The framework rests on a single organising principle: an object of maximal structural indetermination is the categorical limit of every structured object, a statement made precise in the language of complete metric spaces, Hausdorff convergence, and functor theory. The core new contributions, all proved at the level of complete theorems, are: (I) Measure construction (Theorem 4.3): For any IFS {f₁, …, f_N} of contractions on a complete separable metric space X satisfying the open set condition (OSC), and any probability weight vector p, there exists a unique Borel probability measure μ_p on X satisfying the Hutchinson self-similarity equation: μ_p = Σᵢ₌₁N p_i (μ_p ∘ f_i⁻¹) Moreover, μ_p is supported on the IFS attractor A and is atomless when the contraction ratios are irrational. (II) Self-adjointness theorem (Theorem 5.4): The transfer operator H_b : L²(A, μ_p) → L²(A, μ_p) defined by (H_b φ)(x) = Σᵢ₌₁N √p_i · φ(f_i(x)) is a bounded, self-adjoint contraction under OSC. (III) Schrödinger derivation theorem (Theorem 6.2): Stone’s theorem applied to H_b yields, without any external physical input, the Schrödinger equation iℏ d/dt |ψ(t)⟩ = H_b |ψ(t)⟩ as the generator equation of the unitary group e-i H_b t / ℏ on L²(A, μ_p). (IV) Potentiality functor (Theorem 3.7): The five-pillar category USP and the functor P : USP → Phys are rigorously constructed. The functor P is well-defined and preserves the categorical structure of physical limiting maps. We further establish the Arithmetic Potentiality Conjecture (Conjecture 10.6): The K_i functional of the Ortiz Programme (UNASAM-RH-2026) is a USP potentiality functional on the prime lattice, and the Riemann Hypothesis is equivalent to the statement that the zero locus of this functional coincides with the limit class of the prime potentiality space. All results distinguish rigorously between proved theorems, partial results with stated gaps, and open conjectures. No claim labelled “Theorem” contains an unresolved step.