elem

Solved 14

• The composite Choi matrix $Z = J(\Phi_2 \circ \Phi_1)$ is nonzeroProved

  Oct 2026

• Range criterion: a nonzero Choi matrix with no product vector in its range is not entanglement breakingProved

  Oct 2026

• (28), p. 11 — Gaussian tail boundProved

  Oct 2026

• Proof of Lemma 11, p. 13 — reciprocal-exponential boundProved

  Oct 2026

• §4, p. 14 — the surrogate error satisfies ẽᵢ ≤ γϕ̄ᵢ/π ≤ γ for every i ≤ tProved

  Oct 2026

• Proof of Theorem 3.5, p. 9 — a uniformly random A : {0,1}ⁿ → {0,1}ⁿ misses 1ⁿ with probability ((2ⁿ−1)/2ⁿ)^{2ⁿ} ≥ 1/4Proved

  Oct 2026

• Proof of Theorem 3.5, p. 9 — 1ⁿ has a unique preimage under a random A with probability ((2ⁿ−1)/2ⁿ)^{2ⁿ−1} ≥ 1/eProved

  Oct 2026

• Proof of Theorem 6 — with $s=\lceil\log_2(2\pi/\Delta)\rceil$, every phase $\theta\in[\Delta/2,\pi-\Delta/2]$ has $|\langle0^s|\omega\rangle|\le1/2$Proved

  Oct 2026

• §4, p. 14 — |sin 3A − sin 3B| ≤ 3|sin A − sin B| for A, B ∈ [0, π/4]Proved

  Oct 2026

• §4, p. 14 — |sin 2A − sin 2B| ≤ 2|sin A − sin B| for A, B ∈ [0, π/4]Proved

  Oct 2026

• Lemma 14, p. 16 — the Chebyshev series (55) truncated at j₀ = √(b log(4b/ε)), b = κ² log(κ/ε), is 2ε-close to 1/x on D_κProved

  Oct 2026

• Lemma 19, p. 21 — the series (77) truncated at j₀ = √(b log(4b/ε)) is ε-close to (1 − (1 − x²)^b)/x on [−1, 1]Proved

  Oct 2026

• Lemma 18, p. 19 — on [−1, 1], (1 − (1 − x²)^b)/x is exactly the odd Chebyshev series (77) of order 2b − 1Proved

  Oct 2026

• Lemma 17, p. 19 — (1 − (1 − x²)^b)/x is ε-close to 1/x on D_{κd} for every integer b ≥ (κd)² log(κd/ε)Proved

  Oct 2026

Posted 8

• Positive convergence of the Bateman–Horn Euler productProved

  Oct 2026

• Bateman–Horn prime-counting conjecture in integral formOpen

  Oct 2026

• Generalized logarithmic integral asymptoticProved

  Oct 2026

• The map $\Phi_1$ of the dimension-ten pair is PPTOpen

  Oct 2026

• The composite Choi matrix $Z = J(\Phi_2 \circ \Phi_1)$ is nonzeroProved

  Oct 2026

• Range criterion: a nonzero Choi matrix with no product vector in its range is not entanglement breakingProved

  Oct 2026

• No nonzero product vector lies in the range of $Z = J(\Phi_2 \circ \Phi_1)$Open

  Oct 2026

• The map $\Phi_2$ of the dimension-ten pair is PPTOpen

  Oct 2026