[{"type":"text","text":"Task 3.8 is complete and committed (`fa5a74c`). Here's the outcome:\n\n## R-004 — Explicit finite-height Li theorem (ledger #6)\n\n**λ_n > 0 for all 1 ≤ n ≤ N₁ = (T₀−1)² = 8 999 999 999 994 000 000 000 001 ≈ 9.0×10²⁴**, with T₀ = 3×10¹² (Platt–Trudgian 2021). Tag: [PROVED-CONDITIONAL(computational RH-to-height-T₀)]. This is the first fully assembled explicit chain of its kind — Lagarias had the skeleton with untraced O-constants; by Voros's sharpening it's the *maximal* content of the Li route short of proving RH.\n\n**The chain** (D-037):\n- **S_∞ lower bound** (new, elementary): S_∞(n) ≥ ½n log n + (C₁−γ/2)n − 4 via finite differences from Lagarias (4.11) — avoids tracing his §5 lemmas entirely. Validated against E-021's exact values.\n- **|S_f| ≤ B(n)** with every constant traced (Trudgian N(T)/S(T), kernel bounds, R-bound, zero-avoidance).\n- **Margin convexity**: margin(n) > 0 and increasing from N₀ = 5×10⁴ (margin 67,705) all the way to N₁ (margin 2.5×10²⁶).\n- **Finite verification**: ours to n=3000 + Misguich–Voros to 5×10⁵, overlapping N₀ 10×.\n\n**Finds along the way** (the value of doing the assembly):\n- A genuine **gap in D-036**: the uniform ζ′/ζ bound on the left vertical was only verified for |t| ≥ 10, but the line passes within ½ of the trivial pole at w = −2. Closed by direct computation: M₀ = 1.3352 (tame).\n- Lagarias's printed bridge window T²/(4log²T) is slack vs his own proof — we get the sharper (T₀−1)², with the conservative N₁* = 2.7×10²¹ documented as fallback.\n- Adversarial inventory P1–P6/H1–H5 with sensitivity analysis: the theorem is insensitive to its two heuristic constant links (upgrade paths D-039, E-023b listed, not required).\n\n**Phase 3 (Track C) is now fully closed.** Next per plan: task 2.10 (E-022 spectral-measure numerics, Front B), then 2.9-D1 and 2.11. The stray duplicate compendium draft in `research/paper/` is still flagged for your call."}]