The Two-Cone Toeplitz Current Is Relative Flux, Not Unsigned Density
For the bilateral valuation basis, positive and negative cones share the
vacuum and are exchanged by reflection. Their depth-k unilateral defect
projections are both positive and both have rank k.
Three combinations differ:
- direct-sum density has rank
2k; - their union identifies the vacuum and has rank
2k-1; - oriented relative flux is the anti-diagonal pair
(P_k^+,-P_k^-)and has zero total index without being zero as relative data.
The Euler logarithmic derivative selects the last type because scale reversal changes the sign of a derivative. Fourier therefore reverses the current’s orientation, not the positivity of either defect projection.
The common vacuum cancels in the oriented difference. It is seam incidence; the depth-two and higher cells remain on the two reduced arms. The outstanding RH gate is the archimedean pairing of this relative flux.
Research packet: the-two-cone-toeplitz-current-is-relative-flux-not-unsigned-density.md
Graph admission: ev-000000008430-3b275b05-2588-44f4-a945-31c0107d9ef6.