D-002 — Anthropic paper: statements & notation map

Date: 2026-08-17 | Session: 001 | Task: 0.2a | Status: complete Source: research/paper/anthropic-paper-fulltext.txt (17 pp., extracted via pypdf)

1. THEOREM STATEMENTS (verbatim content, re-typeset)

Zeros: ρ = β + iγ, multiplicity m_ρ ≥ 1. Counting functions on 0 ≤ T₁ < T₂:

Chain of trivial inequalities: N^s_0 ≤ N*_0 ≤ N_0 ≤ N and N^s_0 ≤ N^s ≤ N_d ≤ N.

Riemann–von Mangoldt: N(T,2T) = (T/2π)(log(T/2π) + 2 log 2) − T/2π + O(log T).

Theorem A. As T→∞: (i) N^s_0(T,2T) ≥ (2/3 − o(1)) N(T,2T) [simple & on-line] (ii) N_d(T,2T) ≥ (5/6 − o(1)) N(T,2T) [distinct] With Montgomery–Taylor window ψ_MT: constants become 2 − c⁻¹_MT = 0.67250… and ½(3 − c⁻¹_MT) = 0.83625…, where c⁻¹_MT := ½ + (1/√2) cot(1/√2). ψ_MT is optimal among windows ψ (certificates 2−R(ψ)) [CCLM17, Cor. 14]. Ceiling over ALL bandwidth-one certificates ≈ 0.682 (§7.2). A fortiori same bound for N*_0, N_0, N^s. Also holds for (0,T) with rate O(log log T / log T) (Remark 6.1). Prior unconditional records: 5/12 [PRZZ20], 0.6603 [Wu15]. Under RH: 0.6792 via SDP with form-factor positivity outside [−1,1] [CGdL20].

Theorem B. Theorem A verbatim for L(s,χ), χ any fixed primitive Dirichlet character.

Arithmetic inputs (complete list): Weil explicit formula; Riemann–von Mangoldt + N(t,t+1) ≪ log t; Stirling for Γ′/Γ; Chebyshev–Mertens for Σ_{n≤X} Λ(n)² and Σ Λ(n)²/n; Montgomery–Vaughan inequality for {log n: n≤X}, X ≤ T. NO mollifier, zero-density estimate, or zero-free region.

2. PROOF SKELETON (chain 1.2 — the spine to reproduce in 0.2b–0.2d)

Weil form W(f,g) = Σ_ρ m_ρ f̂(γ_ρ) ḡ(γ_ρ), γ_ρ = (ρ − ½)/i. Restrict to d ~ N(T,2T) modulated copies of window ψ, equispaced in [T,2T] → d×d real symmetric matrix G̃, normalized so an isolated simple on-line zero contributes 1 to tr G̃.

• (Z) G̃ = P + Q up to o(1) trace-norm tail. On-line distinct zeros → rank-one positive forms (P); off-line pairs {ρ,1−ρ̄} → signature (1,1) blocks (Q). With s₁,s₂,p = #(simple on-line, multiple on-line points, off-line pairs): tr P ≤ N_0, n₊(Q) ≤ p, N ≥ s₁ + 2s₂ + 2p, tr G̃ = (1+o(1))N. (Propositions 4.1–4.3)
• (P) ‖G̃‖²_HS = (R(ψ) + o(1))N (Lemma 5.6). R(ψ₀) = 4/3; R(ψ_MT) = c⁻¹_MT. = Montgomery unconditional second moment [Mon73, Ary22, BGSTB24] (Theorem 5.7).
• (L) Rank–trace inequality (Lemma 3.2): for Hermitian P₁ ⪰ 0 and Q′ with n₊(Q′) ≤ b, rank P₁ ≥ 2 tr P₁ + 4 tr Q′ − 4b − ‖P₁+Q′‖²_HS. (1.1)

Main chain (1.2): with P₁ = simple-on-line part of P, Q′ := G̃ − P₁: N^s_0 + o(N) ≥ rank P₁ ≥ 4 tr G̃ − 2N − ‖G̃‖²_HS = (2 − R(ψ) − o(1)) N. First ineq = Prop 4.1; second = (L) + tr P₁ + 2n₊(Q′) ≤ N from (Z); equality = (Z),(P). ψ=ψ₀ → Thm A(i); ψ=ψ_MT → 0.6725. For (ii): rearranging → 3s₁ + 4(s₂+p) ≥ (4−R(ψ))N; subtract s₁ + 2s₂ + 2p ≤ N → 2(s₁+s₂+p) ≥ (3−R(ψ))N → N_d ≥ s₁+s₂+p ≥ ½(3−R(ψ)−o(1))N.

(1.1) is the matrix form of m² ≥ 2m−1; with simple zeros on the rank side and multiple ones at flat charge 4 it recovers m² ≥ 3m−2.

3. NOTATION & DEFINITIONS MAP (§1.7, §2)

• f̂(ξ) = ∫_ℝ f(u) e^{−iuξ} du (unitary-less Fourier transform).
• n₊(R) = # strictly positive eigenvalues; ‖R‖²_HS = tr R².
• ℓ := log(T/2π); D₀ := T^{1/2}; ℓ₁ := ℓ + 2 log 2 − 1.
• Explicit-formula components (2.1–2.2): μ(τ) := (1/2π) Re (Γ′/Γ)(1/4 + iτ/2) − (log π)/2π [smooth archimedean density] Π_X(τ) := (1/π) Re X^{1/2+iτ}/(1/2+iτ) [pole absorber] P_X(τ) := −(1/π) Σ_{n≤X} Λ(n)/√n · cos(τ log n) [prime side] ν_X := μ + Π_X + P_X. Weil (2.1): Σ_ρ m_ρ F̂(γ_ρ) = F̂(i/2) + F̂(−i/2) + ∫ F̂ μ dτ − 2 Σ_n Λ(n)/√n F(log n), F even ∈ C²_c(ℝ). If supp F ⊂ [−log X, log X]: poles absorbed → Σ_ρ m_ρ F̂(γ_ρ) = ∫ F̂(τ) ν_X(τ) dτ. (2.2)
• Bounds (2.3–2.5): μ even smooth increasing in |τ|, μ ≥ μ(0) > −1, μ(τ) = (1/2π) log(|τ|/2π) + O(τ⁻²), μ′ ≪ |τ|⁻¹; |Π_X(τ)| ≤ 3√X/(1+|τ|); |P_X(τ)| ≪ √X; ∫_T^{2T} μ = Tℓ₁/2π + O(1/T) = N(T,2T) + O(ℓ); ∫_T^{2T} μ² = Tℓ₁²/4π² (1 + O(ℓ⁻²)).
• Windows (2.6): ψ₀ := 1_[−1/2,1/2]; ψ_MT(s) := cos(√2 s)·1_−1/2,1/2.
• Test function (2.7): χ ∈ C^∞ nondecreasing, χ|(−∞,0]=0, χ|[1,∞)=1; L := ℓ, X := e^L = T/(2π); ϕ(u) := χ(L/2 + u) χ(L/2 − u) · ψ(u/L)^{1/2}. ϕ even, 0≤ϕ≤1, supp ϕ = [−L/2, L/2]; ϕ² = ψ(u/L) off O_χ(1) transition intervals; supp(ϕ∗ϕ) ⊂ [−L,L] = [−log X, log X] → (2.2) applies.
• Paley–Wiener (2.8): |ϕ̂(z)| ≪_χ e^{(L/2)|Im z|} min(L, |z|⁻¹, |z|⁻²). C_χ := ‖(ϕ²)″‖₁ ≪_χ 1.
• Grid: α_k := T + 2πk/L, k ∈ ℤ; d := ⌊LT/(2π)⌋ so α_0..α_{d−1} ∈ [T,2T), d = N(T,2T) + O(L).
• Zero vectors (2.9): v_ρ := (ϕ̂(γ_ρ − α_k))_{0≤k<d} ∈ ℂ^d.
• Matrices (2.10): I := [T,2T), I′ := [T−√T, 2T+√T); zeros with Re γ_ρ ∈ I′ split into on (β=½) / off (β≠½). a := ‖ϕ‖²₂/L = ∫ψ + O_χ(L⁻¹). G̃ := (aL²)⁻¹ Σ_{Re γ_ρ∈I′} m_ρ v_ρ v_ρᵀ; P := (aL²)⁻¹ Σ_{ρ∈on} m_ρ v_ρ v_ρᵀ; Q := G̃ − P; Ẽ := (aL²)⁻¹ Σ_{Re γ_ρ∉I′} m_ρ v_ρ v_ρᵀ. Functional equation pairs {ρ, 1−ρ̄} with v_{1−ρ̄} = v̄_ρ → G̃, Ẽ real symmetric.
• (2.11): (G̃ + Ẽ){kk′} = (aL²)⁻¹ ∫_ℝ ϕ̂(τ−α_k) ϕ̂(τ−α{k′}) ν_X(τ) dτ.

4. KEY LEMMAS ALREADY SEEN (pp. 1–5)

• Lemma 2.1 (Poisson–Gabor identity). For all z,z′ ∈ ℂ: Σ_{k∈ℤ} ϕ̂(z−α_k) ϕ̂(z′−α_k) = L·ϕ²̂(z−z′); in particular Σ_k ϕ̂(z−α_k)² = L‖ϕ‖²₂ = aL². For z=z′ real, truncating: ‖v_ρ‖²₂ ≤ aL². Proof: Poisson summation on Υ(s) = ϕ̂(z−s)ϕ̂(z′−s); dual lattice of h = 2π/L is Lℤ; only m=0 term survives (CRITICAL DENSITY — no aliasing). ⇒ translation-invariant frame kernel at critical density for any window supported in length L. ← the uncertainty-principle anchor (D-001 Q6).
• Lemma 2.2 (Montgomery–Vaughan bilinear form). Distinct λ_r ∈ ℝ, δ_r := min_{s≠r}|λ_r−λ_s|: |Σ_{r≠s} x_r z_s/(λ_r−λ_s)| ≤ (3π/2)(Σ|x_r|²/δ_r)^{1/2}(Σ|z_r|²/δ_r)^{1/2}. Applied with λ = {log n : n ≤ X prime power}; δₙ⁻¹ ≤ 2n (2.12).

5. SECTION MAP FOR REMAINING SUB-STAGES

6. FIRST-PASS OBSERVATIONS (adversarial pre-notes)

1. The whole proof needs only tr G̃, ‖G̃‖²_HS, and block inertia — a 3-number certificate. §7.2 sharpness applies to THIS data; escaping the ceiling needs genuinely new data (higher moments or support > 1).
2. X = T/(2π) exactly at the Montgomery–Vaughan edge X ≤ T; the diagonal domination on the prime side is what forces bandwidth one.
3. The normalization a = ∫ψ makes tr G̃ ≈ N exact by Lemma 2.1 — the critical-density frame is doing the bookkeeping that RH used to do.
4. Davenport–Heilbronn/Epstein insensitivity confirmed in §1.4: the method cannot see o(N) off-line zeros → any claimed RH proof must use DIFFERENT input. Sanity gate stays active.