Part VII: Omega-Germs & the Algebraic Lemniscate
The Prime Polarity Theorem (Part VI) established that every internal prime carries a canonical polarity: B-dominant or C-dominant. Both classes are infinite. This is a finite-regime result — each prime’s polarity is decidable in finite time.
This Part passes from the finite regime to the infinite limit. The central concept is the omega-germ: a compatible tower on the primorial ladder, the τ-native analogue of a Cauchy sequence. Omega-germs live on the bare-metal ontic elements of τ-Idx — they require no coordinates, no interior points, no imported topology. They are pre-topological boundary data.
The bipolar partition of the primes induces a partition of the omega-germs into polarized families: B-polarized germs (where the η-channel freezes) and C-polarized germs (where the γ-channel freezes). The question then becomes: what is the natural scalar algebra for functions on this boundary?
The answer is the bipolar spectral algebra H_τ = A_τ^(B) × A_τ^(C): a split-complex scalar ring with j² = +1 and canonical idempotents e_± = (1 ± j)/2. The bipolar prime partition forces split-complex over elliptic scalars — the algebra carries the B/C partition as an intrinsic feature. From this algebra, the algebraic lemniscate 𝕃 emerges as a theorem: two idempotent sectors e_+ H_τ and e_- H_τ form two algebraic “lobes,” meeting at the crossing-point germ. The lemniscate is not postulated but earned — it is the algebraic readout of the polarized boundary data. The geometric form S¹ ∨ S¹ emerges in Book II when topology is earned.
The chapter sequence: the relevant chapter defines omega-germs on the ontic elements. the relevant chapter introduces polarized omega-germs and the crossing-point germ. the relevant chapter discovers the bipolar spectral algebra and defines the algebraic lemniscate. the relevant chapter reflects on the hinge block in retrospect and previews the road ahead.