M25b Algebra of g-ONS: Advanced Topics: The Sub-Addition Interval, ISHE as Fourier-Mellin Boundary, Conditional Riemann Hypothesis from Branch Structure, and Cloned Microdimensions

This monograph extends the g-ONS programme into several advanced directions: the sub-addition interval, ISHE geometry, conditional RH structures, and cloned microdimensions. The interval R in (-inf, 1] is analysed as a continuous operational region interpolating between Zeration and addition. The symmetric diagonal law HRˢym (a, a) = a + aR connects the zerational increment +1 at R = 0 with additive doubling at R = 1. As R -> -inf, the structure approaches an operational Dirac comb linked to pre-initiation. A major contribution is the identification of the Inter-Shelf Half-Etage (ISHE) at HC rank R = 3. 5 as the Fourier-Mellin boundary between the K-shelf and slog-shelf. Fourier analysis governs the TransAlgebra region, while Mellin analysis governs TetraAlgebra; ISHE appears as their natural transition layer. The monograph also studies complex half-ranks such as R = 0. 5 + iT, whose NC structure mixes addition and multiplication through bilinear terms. Further sections investigate conditional RH structures arising from branch geometry and the emergence of cloned microdimensions inside the HC Rank Manifold. ( ( (contact: pawel@garycki. com) ) )