Chapter 42: The Crossing Point and Bipolar Fourier Analysis

The algebraic lemniscate 𝕃 has two sectors (B and C), and the spectral decomposition theorem (the relevant theorem, I.T12) splits every element of 𝕃 into its B-sector and C-sector components. This chapter studies the crossing point: the algebraic locus where the two sectors meet. The crossing point is singular — it is where e_+ and e_- both contribute, and where the sectorial decomposition degenerates. Its local structure reflects the wedge singularity of the lemniscate. We then formalize the bipolar Fourier transform: the map that sends an element of 𝕃 to its pair of character evaluations. This transform is the formal framework for harmonic analysis on 𝕃, and it previews the central role that lemniscate characters will play in the Central Theorem of Book II: O(τ³) ≅ A_spec(𝕃).