What question did this study set out to answer?

This work aims to formulate the mathematical structure of a new family of operator algebras based on the roughness index alpha.

April 27, 2026Open Access

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

Key Points

This work aims to formulate the mathematical structure of a new family of operator algebras based on the roughness index alpha.
Introduced a one-parameter family of operator algebras governed by the roughness index alpha.
Defined Fractional Hilbert Space and Sunggil Fractional Derivative.
Derived mathematical relationships including the Deformed Heisenberg Algebra and Generalized Weyl Relations.
Defined the Fractional Hilbert Space mathcal{H}_alpha and established the Sunggil Fractional Derivative.
Proved the Deformed Heisenberg Algebra relationship: [hat{x}, hat{p}_alpha] = i hbar (1/alpha) mathbf{1}.
Verified the transformation from Heyting algebra to Boolean algebra as alpha approaches 1.

Abstract

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 mathcalHₐlpha and Sunggil Fractional Derivative. 2. Proof of the Deformed Heisenberg Algebra: hatx, hatpₐlpha = i hbar (1/alpha) mathbf1. 3. Derivation of Generalized Weyl Relations. 4. Verification of the Categorical Logic Phase Transition from Heyting algebra to Boolean algebra as alpha->1.

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

Key Points

Abstract

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