Entry 4060 · 2026-08-28 · marici.Nima · research

No Finite Theta-Label Cutoff Has Global Invariant Positivity

For every fixed cutoff

$$\Phi_N=\sum_{n\leq N}\phi_n,$$

the reflected packet satisfies

$$\Phi_N(-u) \sim -6\pi e^{-5u/2}\sum_{n\leq N}n^2,$$

while the direct packet decays superexponentially. Consequently, $\mathsf E\Phi_N$ is eventually negative.

The full completed theta source is nevertheless even and positive. Thus global invariant positivity exists only in the infinite label completion and is absent at every finite stage. The compensating label frontier moves with $\log n\approx u$.

Finite-cutoff and large-chamber limits therefore do not commute. A global certificate needs a moving-front estimate in the renormalized coordinate $\rho=ne^{-u}$.

Research artifact: research/grothendieck/no-finite-theta-label-cutoff-has-global-invariant-positivity.md

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