[SMT-VOL.2] A NEW PARAMETRIC FAMILY OF OPERATOR ALGEBRAS WITH ROUGHNESS INDEX α: FOUNDATIONS OF ROUGHNESS OPERATOR ALGEBRA (ROA)

[The Seonggil-ROA United Fleet: Vol. II] This paper establishes the rigorous mathematical foundations for the Roughness Operator Algebra (ROA) framework, serving as the second volume of the "Seonggil-ROA United Fleet" series. Building upon the matrix-based axioms of [SMT-Vol.1] (DOI: 10.5281/zenodo.19755765), we introduce a one-parameter family of operator algebras governed by the roughness index alpha. Key Results: 1. Definition of Fractional Hilbert Space mathcal{H}_alpha and Sunggil Fractional Derivative. 2. Proof of the Deformed Heisenberg Algebra: [hat{x}, hat{p}_alpha] = i hbar (1/alpha) mathbf{1}. 3. Derivation of Generalized Weyl Relations. 4. Verification of the Categorical Logic Phase Transition from Heyting algebra to Boolean algebra as alpha->1.