D-033 — Task 3.5b: the Li asymptotic attack — theorem skeleton, constant-tracing plan, and the exact wall

Date: 2026-08-18 | Session: 006 cont.7 | Task: 3.5 Status: complete (design). Deliverable identified: EXPLICIT finite-height theorem (Phase-3.5 remaining work = constant tracing, D-036/E-021).

1. What the attack can and cannot achieve

Voros (D-032 §2): "λ_n > 0 for all n ≥ N₀" ⟺ RH, for ANY N₀ — the asymptotic attack cannot yield a partial result of the form originally envisioned ("prove λ_n > 0 for all large n"). This is not a defeat but a reframing: the attack's achievable deliverable is

THEOREM-TARGET (explicit finite-height Li bound): There exists an EXPLICIT constant N₁ (expected ~ 10²⁰–10²⁴) such that λ_n > 0 for all 1 ≤ n ≤ N₁, proved from: (a) Lagarias's unconditional structure (math/0404394, Thms 5.1/6.1); (b) RH verified to height T₀ (Platt 2017/2022: T₀ ≈ 3.0×10¹²); (c) fully traced constants for ζ (this project); (d) direct verification λ_n > 0 for n ≤ N₀ (E-011/E-015 machinery + literature to 5×10⁵, Misguich/Voros).

This WOULD be a new explicit result (nobody has assembled the chain), at the criterion-3/4 level — and it is the maximal content of the Li route short of proving RH.

2. The skeleton (all from Lagarias, with constants to trace)

λ_n = S_∞(n) − S_f(n) + 1 [BL Thm 2] = (1/2)n log n + C₁n + O(1) − [λ_n(√n) + O(√n log n)] + 1 with C₁ = (γ − 1 − log 2π)/2 (ζ). For n ≤ (T₀ − 1)²: every zero in λ_n(√n) is on the line ⟹ |λ_n(√n)| ≤ 2N(√n) ≤ explicit N(T) bound (Backlund–Rosser/Trudgian |S(T)| ≤ 0.137 log T + 0.443 log log T + 1.588). Hence λ_n ≥ (1/2)n log n + C₁n − c₀ − 2N(√n) − (C₁′√n log n + C₂′√n

C₃′ log²n + C₄′) — positive for n ≥ N₀(constants) since (1/2)n log n dominates √n·(polylog). Then finite verification n < N₀.

3. The exact wall (why "all n" is out of reach here)

Two equivalent formulations, both mapped:

Zero side (Voros): off-line zeros with β < 1/2 give terms |1−1/ρ| > 1, exponentially growing oscillations (non-tempered). Zero-density + height-verification bound their contribution only up to n ~ T₀²·polylog — beyond that the oscillations are uncontrolled.
Arithmetic side (triangle-inequality explosion): bounding Σ_j C(n,j)η_{j−1} termwise gives ~σ₂·14·(15/14)ⁿ — exponential. The true sum has a THREE-WAY cancellation (η-sum, ζ(j)-sum, Stieltjes-sum): η_j ≈ (−i)^{j−1}σ'_j with σ'_j ~ 14^{−j} (same phase for all large-γ zeros — no cancellation inside η_j), so Σ C(n,j)η_j ~ c(1 − i/14)ⁿ, modulus ~ e^{n/392} — the measured λ_n ~ (n/2)log n exists only because the other sums cancel this exponential. Controlling the joint cancellation unconditionally = controlling the zero-side non-temperedness = RH. LOGGED as the precise obstruction; do not retry termwise bounds.

4. Constant-tracing work plan (next sub-stages; each committable)

D-036 (analysis): explicit |ξ′/ξ| and |L′/L| bounds on the Lagarias contour for ζ (Lemma 6.1 with constants; partial-fraction for ξ′/ξ; Davenport Ch. 15 constants). |k_n(s)| segment bounds are already explicit in his proof (e⁴+1, e, 2).
E-021 (numerics): S_∞(n) exact computation for n ≤ 10⁴ (his T-function decomposition — closed forms exist) → explicit O(1) constant; also measures the true |S_f(n)| curve (Maślanka: < 20 at n ≤ 3300) to benchmark the traced bound.
D-037 (assembly): the full chain → N₁ explicit; finite verification table (E-011/E-015 to 3000 + literature to 5×10⁵ + E-021 extension if N₀ > 3000); adversarial review (the N₀ must not rely on unproven inputs; Platt's height T₀ is computational-fact input — tag [PROVED-CONDITIONAL(computational RH-to-height-T₀)]).
Optional: promote to R-004 + paper numerics appendix.

5. Honesty notes

The theorem-target depends on Platt's computational verification of RH to T₀ — an accepted computational fact (interval-arithmetic based), tagged as such.
Expected N₁: (T₀−1)² ≈ 9×10²⁴ capped down by the traced constants' N₀ and the n ≤ T²/(4 log² T) window ≈ 10²² — final value only after D-036/E-021. Do NOT quote a number before the constants exist.
This task does NOT weaken the Voros wall: "all n" stays ⟺ RH.