Research proof index

Let $Z(s)$ be the project's entire zeta surrogate, obtained by removing the pole of $\zeta(s)$ at $s=1$. There is a constant $C>0$ such that, for every $R\ge1$, the total multiplicity of the zeros of $Z$ in $|s|\le R$ is bounded by $C(1+R)^{3/2}.$

Lp limit uniqueness: if f_n → g₁ and f_n → g₂ in eLpNorm, then g₁ =ᵐ g₂.

Vector eLpNorm ≤ sum of component eLpNorms for Pi-valued functions. Uses eLpNorm_mono_real for the pointwise bound together with eLpNorm_sum_le for ℝ-valued functions, avoiding Pi instance synthesis.

If f n → g in eLpNorm and each f n ∈ Lp, then g ∈ Lp, provided g is AEStronglyMeasurable. Avoids the Lp type entirely. The key observation: eLpNorm (f n - g) → 0 means eLpNorm (f N - g) < 1 for some N. Then eLpNorm g ≤ eLpNorm (f N - g) + eLpNorm (f N) < ∞.

Scalar Cauchy → limit. Generic over codomain E and domain α.

Vector eLpNorm convergence → component eLpNorm convergence. Uses eLpNorm_pi_component_le via an explicit function equality to avoid expensive defeq checks.