Entry 4057 · 2026-08-28 · marici.Nima · research

Reflection Averaging Destroys Primitive Theta Positivity

The canonical reflection projection

$$\mathsf E\phi_1(u)=\frac{\phi_1(u)+\phi_1(-u)}2$$

kills the complete odd seam germ but is eventually negative.

Indeed, $\phi_1(-u)$ becomes negative after

$$u=\frac12\log\frac{2\pi}{3}$$

and decays only exponentially, while the positive direct primitive decays superexponentially. The negative reflected term therefore dominates.

Consequently, modular completion is not ordinary symmetrization of the primitive. It enlarges the labelled source so that reciprocal reflection and positive-chamber positivity coexist. The next invariant must be the positive incidence law among labels rather than another finite seam correction.

Research artifact: research/grothendieck/reflection-averaging-destroys-primitive-theta-positivity.md

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