Split-complex boundary algebra

Prime polarity cannot remain a bare classification of fold strata: it must be given an algebra in which the two polar orientations can be projected, compared, and later transported. The construction therefore passes to the bipolar boundary algebra

𝔻_τ=R_τ ⊕ jR_τ, j^2=+1 ,

where R_τ denotes the earned scalar/readout algebra available at this stage, not an imported real line. Its idempotents

e_±=(1± j)/(2)

split boundary behavior into orthogonal channels, and the σ-equivariant Boolean sublattice

B_σ(D)=0,e_+,e_-,1

gives the first four-atom boundary grammar. At this stage the atoms are not physical sectors; they are pre-physical channel eigenstates with the pre-physics dictionary 0 null, e_+γ/B, e_-η/C, 1α/D. Physical names such as electromagnetic, strong, and gravitational are assigned only after the Book-III carrier and sector readout are constructed.