R-001 — Li coefficients λ_n for n ≤ 2000 from the arithmetic (zero-free) side

Date: 2026-08-17 | Session: 004 | Track: C (task 3.2 continuation) | Tag: [NUMERICAL-VALIDATED]

Statement

λ_n computed for 1 ≤ n ≤ 2000 entirely from arithmetic data — Stieltjes constants γ_k (flint.acb.stieltjes, rigorous Arb balls) and ζ(j) (flint.arb) via Bombieri–Lagarias Theorem 2 (J. Number Theory 77 (1999) 274–287), η-form: λ_n = 1 − Σ_{j=1}^n C(n,j) η_{j−1} − (log 4π + γ)n/2 − Σ_{j=2}^n (−1)^{j−1} C(n,j) (1−2^{−j}) ζ(j). No zeros used. All λ_n > 0 for n ≤ 2000 (consistent with Li's criterion ⟺ RH).

Proof / Derivation

Numerical, dps = 1000 (budget 0.45·n_max + 100 for the 2^n and (4/3)^n cancellations). Validations (E-011, all PASS):

V1: λ₁ err 4.31e-102 vs 1 + γ/2 − log(4π)/2 (flint, 100-digit reference).
V2: n ≤ 116 cross-check vs E-008 zero-side table within tolerance (worst dev ratio 0.125 of allowed max(2·tail, 2% rel)).
V3: λ_n > 0 for all n ≤ 2000.
V5: self-consistency at 1.4× dps on {1,2,100,1000}: rel dev 6.67e-705.
Asymptotic: λ_n / [(n/2)(log n − log 2π + γ − 1)] = 1.00216 at n=2000 (Arias de Reyna refined law, →1). Key values: λ₁₀₀₀ = 2326.0532, λ₂₀₀₀ = 5351.7595.

Dependencies

E-008 (zero-side cross-check), D-001 NT-1 (explicit-formula conventions), session-003 env finding (mpmath 1.3 + Py 3.14: mp.zeta broken >16 digits → all ζ/Stieltjes via flint; mp.mp.dps setter, not mp.dps).

Adversarial review

mpmath arithmetic is not interval-rigorous; certification rests on V5 self-consistency + V2 zero-side cross-check. Arb ball-arithmetic upgrade would make this fully rigorous (queued as possible hardening).
Session-004 incident: an orphaned run (older script revision, killed mid-edit) produced V1–V3 failures with a 53-bit signature; also the original hardcoded λ₁ constant was wrong from digit 22 (correct value ...03381431..., flint-verified). Both fixed; final run clean.
D-H/Epstein gate: not applicable (certificate table, not a criterion claim).

Paper section