Two-Sided Closure Theory: A Self-Contained Introduction via the Koide Formula

Two-Sided Closure Theory (TSCT) is a research programme that uses subfactor structure on a two-sided modular pair to constrain Standard Model parameters. This paper introduces only the minimum machinery needed to identify one specific result - the Koide lepton mass relation QK = 2/3 - as a candidate structural consequence of the A₃ subfactor at Jones index 2. The construction depends on one explicit assumption (a generation-channel correspondence) and one explicit conjecture (a saturation property of the S₃-symmetric channel weights), both named and labelled as such. Granted the assumption, the conjecture is what would force the amplitude in Foot's parametrisation of Koide's relation to take its observed value A² = 2. The paper is deliberately self-contained: no prior familiarity with TSCT is required, no other TSCT work is cited, and the only references are to standard subfactor-theory literature (Jones, Pimsner-Popa, Goodman-de la Harpe-Jones, Takesaki, Evans-Kawahigashi, plus standard flavour-physics references for Koide, Foot, and competing flavour frameworks). A reader familiar with subfactor theory at the level of Jones (1983) and Pimsner-Popa (1988) should be able to assess the proposal independently. The note serves as the recommended entry point for the broader TSCT programme: a reader who finds the conditional construction reasonable may then examine the broader applications (electroweak hierarchy, CKM, PMNS, dark energy) ; a reader who does not is unlikely to find longer expositions more convincing. A precise specialist question on the conjecture L1 is given in the separate "Embedding Required by the Koide Saturation Lemma" note (10. 5281/zenodo. 20209483). Comments, references, and engagement welcome. Contact: david. m. sparks@proton. me