Research proof index

If val z.2 ≤ c for all z in the support of costDist, then expectedCost ≤ c. This is the key bridge from worst-case (support) bounds to expected bounds.

A more precise version of sfHomotopy_in. -/ theorem sfHomotopy_in' {ι} (h₀ : SurroundingFamily g b γ₀ U) (h₁ : SurroundingFamily g b γ₁ U) (τ : ι → ℝ) (x : ι → E) (i : ι) {V : Set E} (hx : x i ∈ V) {t : ℝ} (ht : t ∈ I) {s : ℝ} (h_in₀ : ∀ i, x i ∈ V → ∀ t ∈ I, ∀ (s : ℝ), τ i ≠ 1 → (x i, γ₀ (x i) t s) ∈ Ω) (h_in₁ : ∀ i, x i ∈ V → ∀ t ∈ I, ∀ (s : ℝ), τ i ≠...

Let $E_\pi(x)=|\pi(x)-\operatorname{Li}(x)|/(x/\log x)$. If $E_\pi$ obeys the classical bound $E_\pi(x)\le A\left(\frac{\log x}{R}\right)^B\exp\!\left(-C\sqrt{\frac{\log x}{R}}\right)$ for every $x\ge x_0$, with $A,B,C,R>0$, then beyond any $x_1\ge\max\{x_0,\exp(R(2B/C)^2)\}$ it obeys the uniform numerical bound obtained by evaluating that expression at $x_1$.

Let $E_\psi(x)=|\psi(x)-x|/x$. If $E_\psi$ obeys the classical bound $E_\psi(x)\le A\left(\frac{\log x}{R}\right)^B\exp\!\left(-C\sqrt{\frac{\log x}{R}}\right)$ for every $x\ge x_0$, with $A,B,C,R>0$, then beyond any $x_1\ge\max\{x_0,\exp(R(2B/C)^2)\}$ it obeys the uniform numerical bound obtained by evaluating that expression at $x_1$.

Sub-lemma for Step 1b: the conj of H_hat by σ^{−γ} simplifies to (ρ.M.conj (σ^γ).mat)^{α−1}. This uses σ^{−γ} · σ^γ = identity (on support) to cancel the outer σ^γ factors. -/ theorem H_hat_conj_sigma (hα : 1 < α) (ρ σ : MState d) : let γ := (1 - α) / (2 * α) (H_hat α ρ σ).conj (σ.M ^ (-γ)).mat = (ρ.M.conj (σ.M ^ γ).mat) ^ (α - 1) := by intro γ have hγ :...

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...