M37, U00-08 The Operational Unification Programme

M37 is the entrance document to the Unification Programme, also identified as U00–U08, built on the broader Operational Manifold corpus. The programme’s central object is the logaderivative connection Λ=log⁡f′, whose curvature is the Schwarzian, whose kernel is PSL(2,K), and whose holonomy is the flux κ. Through this projective object, the document presents one framework for reductions across arithmetic, physics, algebra, and dynamics, including the Riemann Hypothesis, Standard-Model/cascade tuning, Navier–Stokes, algebraic solvability beyond radicals, and the three-body problem. Navitagion: Included papers — one sentence each U00 — Introduction to the Unification Programme: Introduces the U01–U08 sequence, gives the required Manifold background, explains the architecture of the programme, and records the status ledger for its major claims. U01 — The Balance Defect: Defines the Balance Defect through the universal NC quadratic and the Halbzeug locus, presenting it as the common skeleton behind several narrowed problems in the corpus. U02 — The Logaderivative Connection: Identifies carry, Virasoro curvature, and TrigCore flux as three avatars of one projective logaderivative connection, with flux interpreted as holonomy. U03 — The Zeta Tuning and the Reality Reduction: Applies the Balance Defect and logaderivative framework to the zeta tuning, reducing the Riemann Hypothesis direction to a reality condition for the warped-prime flow. U04 — The Rigid Regime and the Discriminant Phase: Studies the cascade or rigid regime on the CCC-plane, deriving a closed-form Balance Defect from the root displacement of the universal quadratic and connecting the phase channel to the physics-side tuning. U05 — The Balance Class at the Halbzeug: Extends the Balance Defect into a Balance Class at the Halbzeug, connecting SC/HSC structure with Navier–Stokes, Stokes/Hodge-type formulations, and BSD-related half-rank behaviour. U06 — The Fractional-Height Ladder and the Navier–Stokes Barrier: Places Navier–Stokes regularity inside a fractional-height ladder and formulates the main obstruction as a norm-window or balance barrier between logarithmic and enstrophic regimes. U07 — The Flux-Rationality Criterion and the Modular Reduction of the Seam: Uses flux rationality to distinguish algebraic from transcendental solvability and relates the solvability boundary to the modular reduction of the seam. U08 — The Three-Body Branch-Path Group and the Dynamical Flux: Applies the flux framework to the three-body problem by introducing a branch-path group whose rational or irrational flux marks the integrable/chaotic boundary