Research proof index

Functional equation of $G$ restricted to the imaginary axis.

G / q^(3/2) → 20480 as im(z) → ∞. Here q^(3/2) = exp(2πi · (3/2) · z).

A weighted Hölder lower bound for Fourier energy. If $\Delta$ lies in the $\eta$-large spectrum of $f$, $m\ne0$, and a weight $\nu$ is at least $1$ wherever $f$ is nonzero, then the order-$m$ energy of $\Delta$ weighted by $\widehat\nu$ is at least $|\Delta|^{2m}\eta^{2m}\|f\|_1^2/\|f\|_2^2$.

The product of two Levi-Civita-type symbols is a generalized Kronecker delta: δ^{μ}_{·} · δ^{ν}_{·} = δ^{μ}_{ν}, where each single factor is a Kronecker matrix against the identity. This is the Lean form of ε^{μ₁…μₙ} ε_{ν₁…νₙ} = δ^{μ₁…μₙ}_{ν₁…νₙ}.

Generalized Kronecker delta contraction. Summing a generalizedKroneckerDelta over one shared index appended at the end lowers the rank by one and pulls out a factor of card α - n. This is the reusable combinatorial fact behind all epsilon-epsilon identities.

For every real $x\ge2$, the extended-nonnegative geometric series satisfies

$\sum_{n=0}^{\infty}2^{-n/x}\le2^x.$