The k=2/k=3 Boundary Algebra and the Origin of Standard Model Mixing Parameters in Two-Sided Closure Theory

We identify the missing organising structure of Two-Sided Closure Theory (TSCT): the boundary algebra B₂₃ at the interface between Jones levels k=2 and k=3. This algebra, the centraliser of TL (k=2) in TL (k=3), has total dimension 29, with sub-dimensions 20 (gauge/curvature sector), 9 (colour sector), and EW carrier dimension 13. Every Standard Model parameter associated with mixing, CP violation, dark energy geometry, and quark isospin ratios is a projection of this single object. Specifically: sin²θW = NC/dimEW = 3/13; Ω_Λ = dimgauge/dimₜotal = 20/29; ACP (local) /ACP (global) = dimgauge/dimcolour = 20/9; mᵤ/md = 2NC/dimEW = 6/13. The Cabibbo angle, the √2/13 neutrino prefactor, and the Koide phase correction 9√2/320 follow as further projections. The colour charge NC = 3 is a boundary invariant: NC = (k₃+2) − M: N₊䃒 = 5 − 2 = 3. It is not a free parameter. This reduces eight independently stated Tier-2 identifications to a single algebraic claim: prove dim (B₂₃) = 29 from Jones bimodule theory. That is the central open obligation of this paper. Principle: the Standard Model mixing parameters are interface costs — the algebraic price of rendering two incompatible closure depths (k=2 and k=3) as one coherent physical world.