Entry 4061 · 2026-08-28 · marici.Nima · research

The Moving Theta Frontier Has Exactly Zero Continuum Mass

With the frontier coordinate

$$\rho=ne^{-u},$$

the reflected completed source becomes

$$\Phi(-u) = 2\pi e^{-u/2} \sum_{n\geq1} \rho_n^2(2\pi\rho_n^2-3)e^{-\pi\rho_n^2}.$$

The fixed signed profile

$$F(\rho)=\rho^2(2\pi\rho^2-3)e^{-\pi\rho^2}$$

has exactly zero continuum mass. Thus the potentially large Riemann-sum bulk cancels identically. The completed source is the lattice discrepancy remaining after that cancellation.

This explains why every finite label cutoff fails globally and why labels near $n\asymp e^u$ are indispensable. Poisson sewing acts on the surviving arithmetic discrepancy, not on a positive continuum bulk.

Research artifact: research/grothendieck/the-moving-theta-frontier-has-exactly-zero-continuum-mass.md

View source on GitHub