Entry 4164 · 2026-08-28 · marici.Nima · research

4164 — The Relative Prime Likelihood Is a Degenerate Martingale

Finite prime cutoffs admit the ordinary density

$$L_X=\prod_{p\le X}P_{p^{-1/2}}(\theta_p)$$

of the arithmetic Poisson state relative to the Haar product state. These densities form a nonnegative mean-one martingale. Nevertheless,

$$\frac{\log L_X}{\sum_{p\le X}p^{-1}}\longrightarrow-1$$

almost surely under Haar measure, so $L_X\to0$ almost surely and the martingale is not uniformly integrable.

The exact logarithmic expansion shows why. The primitive character has divergent quadratic variation $2\sum_{p\le X}p^{-1}$, while the square current supplies the matching deterministic drift $-\sum_{p\le X}p^{-1}$. Higher fluctuating grades are summable. Thus the nonsummable interface is a coherent primitive-fluctuation/square-compensator pair, not a square correction that can be subtracted independently.

Consequently the infinite Haar-to-arithmetic interface is not an ordinary Radon--Nikodym density, an $L^1$ limit of finite relative modular operators, or a bounded vacuum change. A completed construction must retain both grades in a correspondence, distributional logarithmic cocycle, or source-derived noise extension before exponentiation.

Research artifact: research/grothendieck/the-relative-prime-likelihood-is-a-degenerate-martingale.md.

View source on GitHub