G-003 — Density certificates → de Bruijn–Newman Λ

Date: 2026-08-17 | Session: 004 | Track: C (task 3.3)

Attempt

Use the Anthropic paper's unconditional ≥ 2/3 simple-on-line density to constrain the de Bruijn–Newman constant Λ (RH ⟺ Λ = 0 after Rodgers–Tao; best known Λ ≤ 0.2, Platt–Trudgian 2020). Candidate channel: large Λ ⟹ many backward-heat-flow collisions by t = 0 ⟹ large off-line fraction, so bounding the off-line fraction would bound Λ.

Obstruction

One-sided information, three axes (full analysis: D-021):

1. Direction: the theorem LOWER-bounds the on-line fraction (certifies real zeros only); Λ needs an UPPER bound on the complex fraction. No value of Λ contradicts the theorem — the theorem cannot see Λ at all.
2. Extremal vs averaged: Λ is set by the FIRST collision (smallest rescaled gap). E-012: two-body collision times across 10⁴ zeros span 4.5 orders of magnitude (min 1.78e-4 vs median 0.108); full all-to-all ODE on 300 zeros confirms the smallest-gap pair collides first (many- body correction 4.8%). An averaged certificate cannot see the binding pair.
3. Asymptotic vs explicit: Λ ≤ 0.2 stands on RH verified to 3·10¹²; the density theorem is asymptotic in T with no finite-height content.

Lesson

Any route from moment/window density certificates to Λ is closed at the information-type level, not the strength level — do not retry with a stronger density result (even 100% on-line density, e.g. HL*(∞), gives no upper bound on the complex fraction). Λ progress = RH-to-height compute (Platt–Trudgian direction) or a qualitatively new certificate type that certifies zeros as COMPLEX (none known; logged as open problem in D-001 Part III).

Salvage

E-012 retained as the calibrated illustration that Rodgers–Tao two-body collision times are accurate to ~5% against full many-body ODE on the binding window — usable if a future route ever needs collision-time estimates from gap data.