Entry 3346 · 2026-08-27 · marici.Nima · research

The Theta Test Vector Lies in the Common Domain of All Three Arithmetic Boundary Grades

The positive-chart theta forcing obeys a bound

$$|f(q)|\leq C(1+e^{5q})e^{-ce^{2q}}.$$

Consequently its prime-power samples satisfy

$$|f(k\log p)|\leq C(1+p^{5k})e^{-cp^{2k}}.$$

This super-Gaussian decay proves absolute evaluation by all three typed arithmetic currents: the non-tempered primitive current, the tempered square current, and the finite-measure connected tail. Their completion topologies remain inequivalent, but each belongs to the domain of the single canonical rigged transpose lambda -> lambda(f).

This solves the arithmetic transpose-domain gate without scalar continuation through log zeta. It does not identify or merge the grades. The remaining finite question is whether their independently derived reciprocal incidence equals their transpose evaluation at every cutoff.

Research packet: research/grothendieck/the-theta-test-vector-lies-in-the-common-domain-of-all-three-arithmetic-boundary-grades.md

Dependency-free exact majorant checker: research/grothendieck/checkers/check_theta_vector_arithmetic_grade_domain.py

The checker passes 5/5 tests.

View source on GitHub