Research proof index

Step 3 (Convexity in σ): For fixed H ≥ 0 and ρ, and α > 1, the map σ ↦ f_alpha α H ρ σ is convex. The key is that for p = α/(α−1) > 1: • A ↦ Tr[A^p] is convex on PSD matrices (trace function convexity, Theorem 2.10 of Carlen), • σ ↦ σ^{−γ} H σ^{−γ} is concave in σ by Lieb concavity (since −γ = (α−1)/(2α) ∈ (0,½)), • The composition...

Step 1c: H_hat is a maximizer: for all H ≥ 0, f_α(H) ≤ f_α(H_hat). This uses the trace Young inequality: for PSD A, B and conjugate exponents p, q > 1, ⟪A, B⟫ ≤ Tr[A^p]/p + Tr[B^q]/q. Applied with A = σ^γ ρ σ^γ, B = σ^{-γ} H σ^{-γ}, p = α, q = α/(α-1), the inner product identity ⟪ρ, H⟫ = ⟪A, B⟫ (under the support condition) yield...

Reduce a power of I * w via I ^ 4 = 1. -/ lemma I_mul_npow (w : ℂ) (n : ℕ) : (I * w) ^ n = I ^ (n % 4) * w ^ n := by rw [mul_pow, I_pow_eq_pow_mod] /- Functional equation of $F$

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

The vanishing order of F at infinity is 2. Blueprint: F = 720² * q² * (1 + O(q)), so F / q² → 720² as im(z) → ∞.

F₁ has exponential decay at infinity (it's essentially D E₄ which decays).