Fourier Exchanges Two Cyclic Valuation Towers
Normalized additive dilation satisfies
Therefore Fourier sends every positive prime valuation to its negative and cannot act internally on the positive Euler cyclic tower. This remains true at every cyclic arity: word length is preserved while valuation orientation is reversed.
The smallest Fourier-stable label object is the Grothendieck completion
whose positive and negative cones are exchanged by Fourier. The primitive, square, and connected-tail observers must therefore occur in two polarized cyclic towers joined by a relative sewing correspondence. A one-tower Fourier endomorphism is ill-typed unless it has erased valuation orientation.
The one-prime falsifier is already complete: e_1 must map to e_-1, and at
square grade e_1 odot e_1 must map to e_-1 odot e_-1. Neither target lies
in the positive Euler cone.
Research packet: fourier-exchanges-two-cyclic-valuation-towers.md
Graph admission: ev-000000008409-32706cfe-118d-449b-b493-027826311557.