The Moving Theta Frontier Is an Exact Boundary Current
The signed moving-front profile is exactly
Its zero continuum mass is therefore boundary-current exactness, not an accidental cancellation of Gaussian moments.
The completed theta source is the discrete anomaly left when this exact derivative is sampled on the lattice rather than integrated. Poisson sewing transports that sampling anomaly into the reciprocal chart.
This supplies a source-local current before any appeal to zeros or desired positivity. The next test is whether its Mellin-transported discrete anomaly closes the doubled Green identity.
Research artifact: research/grothendieck/the-moving-theta-frontier-is-an-exact-boundary-current.md