D-021 — Task 3.3 scoping: de Bruijn–Newman heat flow vs the 2/3-density

Date: 2026-08-17 | Session: 004 | Fork 3 of 4 (§8.4a) Status: analysis complete — route CLOSED (one-sided information mismatch); toy illustration in E-012.

Question (task 3.3)

Can the Anthropic paper's unconditional result (≥ 2/3 of zeros simple and on the critical line) constrain the de Bruijn–Newman constant Λ?

Background (literature state, verified 2026-08-17; logged D-001 Part III)

• H_t(z) = ∫₀^∞ e^{tu²} Φ(u) cos(zu) du; Λ := inf{t : H_t has only real zeros}.
• RH ⟺ Λ ≤ 0 (de Bruijn 1950: Λ ≤ 1/2; reality propagates forward in t).
• Rodgers–Tao 2018/2020: Λ ≥ 0 (Dobner later re-proof/generalization). Hence RH ⟺ Λ = 0.
• Best upper bound: Λ ≤ 0.2 (Platt–Trudgian 2020, "RH is true up to 3·10¹²"); before that Λ ≤ 0.22 (Polymath15, 2019). No improvement since.
• Upper-bound method: for explicit a, prove ALL zeros of H_a real = (i) RH verified to explicit height + (ii) rigorous heat-flow dynamics estimates (zero ODE + error bounds). Extremal, finite-height, certified.

The candidate channel (and why it fails)

Under the backward heat flow, real zeros of H_t move by dx_j/dt = 2 Σ_{k≠j} 1/(x_j − x_k) (Rodgers–Tao dynamics); a conjugate pair of off-line zeros at t = 0 is born in a collision at some t_c ∈ [0, Λ]. Heuristic: if Λ were large, many collisions would occur by t = 0, forcing a large off-line fraction — so an upper bound on the off-line fraction would bound Λ. The paper supplies the opposite inequality.

Three-axis mismatch:

1. Direction. The 2/3 theorem is a LOWER bound on the on-line fraction (it certifies real zeros; it certifies nothing as complex). Constraining Λ above needs an UPPER bound on the off-line (complex) fraction (off-line ≥ f(Λ)). One-sided information flows the wrong way.
2. Averaged vs extremal. Λ is set by the FIRST collision going backward — governed by the single smallest rescaled gap (Lehmer-pair extremal statistic). The 2/3 certificate is a second-moment/averaged statement; mean gap statistics are independent of the minimum (E-012: collision times across 10⁴ zeros span ≥ 5 orders of magnitude; the binding pair is invisible to any averaged functional we control).
3. Asymptotic vs explicit. Λ ≤ 0.2 stands on RH verified to 3·10¹² (finite, explicit). The 2/3 result is asymptotic in T; no finite-height certificate about any specific zero follows.

Corollary of axis 1: even under RH-false with Λ > 0, the paper's theorem would still hold (2/3 real is compatible with any collision history that leaves ≥ 2/3 density real). So no value of Λ contradicts the theorem — the theorem cannot see Λ at all. This is the same one-sidedness as 3.4a (D-016), one level up: there the certificate couldn't see deep off-line zeros; here it can't see ANY off-line zero.

What WOULD constrain Λ from our side

• Verified zero-free regions / RH-to-height (Platt–Trudgian direction): pure compute + interval arithmetic, no new theory. Not our edge.
• An upper bound on the complex fraction of H_0 zeros (any unconditional statement "≤ c of zeros are off the line") — no known method produces this; the moment/window machinery certifies reality, never complexity. Logging as an open problem in D-001 Part III.

Verdict

3.3 CLOSED: the 2/3 density cannot constrain Λ — not a matter of strength but of direction. Route logged to graveyard (G-003) so it is not retried. Toy illustration E-012 quantifies axis 2 (extremal vs averaged) on our 10⁴-zero table.