Chapter 18: The Internal Bipolar Classifier

The preceding chapters of Part III erected the spectral algebra in stages: the primorial ladder , the CRT decomposition , Hensel lifting , and the adelic embedding . At every stage, an informal vocabulary intruded—”B-lobe dominant,” “C-lobe dominant,” “mixed”—to describe the bipolar character of individual primes. Informal vocabulary is debt. This chapter repays it.

The internal bipolar classifier Label_n is a computable function from primes p ≤ p_n to the label set {B, C, X}, defined entirely from the CRT idempotents on ℤ / Prim(n) ℤ. A prime receives label B when its CRT idempotent projects dominantly onto the χ_+-eigenspace (the exponent stratum of the ABCD chart); label C when the projection is χ_–dominant (the tetration stratum); and label X when the two projections are balanced. The classifier is computable at every finite depth: one evaluates the CRT idempotent, applies the split-complex projection, and reads off the dominant component.

Two structural results close the chapter. First, the classification stabilises: for each prime p, there exists a depth n₀ beyond which Label_n(p) is constant. The limiting classifier Label_∞ is well defined. Second, the label assignment is compatible with the split-complex idempotents e_+ and e_- of the boundary ring H_τ: a B-type prime has e_+-dominant spectral coefficients, a C-type prime has e_–dominant coefficients, and an X-type prime has balanced contributions. A table of explicit computations for the primes 2, 3, 5, 7, 11, 13 at depths n = 1, 2, 3, 4 makes the stabilisation visible.

With the internal bipolar classifier in place, every subsequent chapter can replace informal lobe language with a decidable predicate on primes. The spectral algebra is closed.