D-032 — Task 3.5a: Li-coefficient literature map (what exists, what the target really is)

Date: 2026-08-18 | Session: 006 cont.7 | Task: 3.5 (Li asymptotic attack) Status: complete. Sources recorded per §4A.3.

1. The criterion and the arithmetic form

• Li (1997): λ_n := Σ′ρ [1−(1−1/ρ)^n] > 0 ∀ n ≥ 1 ⟺ RH (conditional convergence; λ₀ = 0, λ{−n} = λ_n).
• Bombieri–Lagarias (1999, JNT 77:274–287): (i) Li criterion for general multisets; (ii) arithmetic form λ_n = S_∞(n) − S_f(n) + 1 (archimedean − finite places + pole term) — the basis of our E-011/E-015 computations; (iii) λ_n = Weil positivity at a specific test function. [local: research/paper/BL-1999-fulltext.txt]
• Our ledger #3: λ_n > 0 verified to n = 3000 (E-011/R-001/E-015), arithmetic side, zero-free.

2. The target hierarchy — Voros sharpenings (KEY structural fact)

• Voros, math/0404213 (Sharpening I): RH ⟺ λ_n ~ n(A log n + B) with explicit A > 0, B. The asymptotic law itself is RH-EQUIVALENT.
• Voros, math/0506326 (Sharpenings II, MPAG 9 (2006)): if RH fails, λ_n eventually takes a non-tempered oscillatory form — sign changes infinitely often. Hence "λ_n > 0 for all n ≥ N₀" is ALSO ⟺ RH, for ANY N₀. There is no partial-strength version of the goal.
• Voros 1602.03292 (simplified variant sequences), 1703.02844 (discretized Keiper/Li; Misguich computed to n = 5·10⁵; tested on Davenport–Heilbronn counterexamples), 2204.01036 (closed forms), 1403.4558 (superzeta/exponential-asymptotic view). Consequence for 3.5: the full asymptotic attack CANNOT yield partial credit — "λ_n > 0 ∀n≥N₀" = RH. The honest deliverable is the FINITE-HEIGHT theorem (§3 below) plus the wall documentation (D-033).

3. The unconditional skeleton — Lagarias (math/0404394, Ann. Inst.

Fourier 57 (2007) 1689–1740) [PDF read directly] - Thm 5.1 (unconditional, archimedean part): S_∞(n) = (N/2)n log n + C₁n + O(1), C₁ = (N/2)(γ − 1 − log 2π) + ½log Q(π). For ζ: C₁ = (γ − 1 − log 2π)/2 — EXACTLY the "Arias de Reyna refined law" our E-011 ratio → 1 measures (ratio 0.999845 at n=3000 ✓ consistent). The O(1) constants are traceable through his Lemmas 5.1/5.2 (absolute O(|z|+1) constants). - Thm 6.1 (unconditional, finite part): S_f(n) = λ_n(√n) + O(√n log n) (contour argument, kernel k_n(s) = (1+1/s)ⁿ − 1; constants traceable: |k_n| bounds e⁴+1, e, 2 on the segments; trivial zeros O(1); left segment O(√n log n) via Lemma 6.1 |L′/L| = O(log(|s|+2))). - (1.16) THE BRIDGE: if RH holds up to height T, then S_f(n) = O(√n log n) for all n ≤ T²/(4(log T)²) — and λ_n(T) = O(T log T) explicitly since each on-line zero contributes ≤ 2 (|1−(1−1/ρ)ⁿ| ≤ 2 when |1−1/ρ| = 1). - Thm 1.1 (assembly): λ_n = (N/2)n log n + C₁n − λ_n(√n) + O(√n log n). - Also: Maślanka (cited): |S_f(n)| < 20 for n ≤ 3300 — the finite part is tiny in practice. Coffey [8,9]: lower bounds for the archimedean contribution.

4. Conditional / numerical landscape

• Mazhouda 1506.01755: under GRH, λ_n = (d/2)n log n + c n + O(√n log n) for Selberg class (saddle-point/Nörlund–Rice).
• Maślanka math/0402168: computed λ_n to 3000+, "strictly growing with tiny oscillations" (numerical). Misguich/Voros: to 5·10⁵.
• Johansson 1309.2877: rigorous Arb computation of Stieltjes/Keiper–Li.
• Maślanka 2211.08993: entire interpolation λ(s); zero patterns (quadruplets vs real doublets) as RH indicator.
• Suzuki 2301.05779: λ_n as model-space norms (reformulation).
• Sekatskii 1304.7895/1305.1421/1404.7276: generalized BL theorem, shift parameter a ≠ 1/2.
• Coffey math-ph/0505052, math-ph/0507042, 0706.0345: bounds on λ_k, constants removing the (1/2)log n leading term, Stieltjes subsum S_γ(n) series.

5. What does NOT exist (the gap our attack addresses)

• No UNCONDITIONAL positivity proof for any infinite range of n (impossible without RH — §2).
• No EXPLICIT version of the finite-height bridge (1.16): Lagarias's implied constants are π-dependent and untraced; nobody has assembled "RH to height T₀ (Platt) + explicit constants ⟹ λ_n > 0 for all n ≤ N₁" with a concrete N₁ and full constant chain. THIS is the achievable deliverable (D-033).
• No unconditional bound on Σ_j C(n,j)η_{j−1} avoiding the triangle-inequality explosion ((15/14)ⁿ; the cancellation IS the content of RH — D-033 §3).