M40a The Symmetric Heun Operation and the Half-Étage Convergence Theorem

M40a develops the Symmetric Heun Operation (SHO) as the HyperCore half-étage operation at rank R=5/2, positioned between multiplication and Cpow (Rank 3 symemtric operaiton), and treats it as the genus-2 successor of the AGM at R=3/2. The paper’s central analytic result is the completion of the SHO convergence theory: the transverse recursion is shown to be quadratically convergent, with an explicit positive quartic correction term that is uniformly bounded for σ≥σ0>0. The main consequence is that the convergence condition (H-Conv) used in the Operational Hodge programme is no longer an open input at codimension p=2, and more generally holds pointwise at every Shell–Thron-admissible codimension. The only remaining uniform-in-p issue is identified with the already existing Propagation Conjecture, which the paper explicitly says is not required for the Hodge application. Thus the Operational Hodge reduction is sharpened: after M40a, the Hodge programme reduces codimension by codimension to the arithmetic condition (H-Tate) alone. The paper also constructs the conserved quantity of the SHO. The longitudinal limit σ∞ to be an exact first integral of the SHO map, and the attractor invariant is identified as K*5/2=L*sqrt(ln(L)), with L=e&(σ∞), equal to the geometric mean of the rank-two and rank-three conserved quantities evaluated at the attractor. Conceptually, M40a places the SHO on the genus–rank ladder R=1+g/2: AGM at R=3/2 corresponds to genus 1 and underlies the operational BSD reduction, SHO at R=5/2 corresponds to genus 2 and supports the Hodge programme, and ISHE at R=7/2 corresponds to genus 3. The paper records a structural thesis for the sequel: the SHO Coulomb-branch monodromy and the BSD Koenigs-monodromy crossing are two manifestations of the same crossing mechanism.