MISSION: Repairing the counterexample — the heat flow clock on Davenport–Heilbronn

Agent Persona: The Hunter (unsupervised-fun edition, second outing) Scope: hunts/flow_repair/ only. Nothing outside this directory is modified except the case-log entry in hunts/README.md.

Objective

The lab's two showpiece arms have never met. zeta/heatflow.py runs the de Bruijn–Newman flow H_t = exp(−t∂²)H₀ on the Riemann Ξ — where increasing t pulls complex zero pairs onto the real axis, Λ is the last landing time, and RH ⟺ Λ = 0. zeta/epstein.py owns the Davenport–Heilbronn function — zeta's functional equation, real coefficients, a real Hardy Z, and zeros off the critical line. The flow has only ever been pointed at the function that (as far as anyone knows) needs no repair.

Point it at the one that provably does. The hunt asks one question:

How long does the backward-heat flow take to repair the Davenport–Heilbronn counterexample — pair by pair, measured, with the collision times read off a quantity that is analytic through the collision?

Each off-line quadruple ±(γ ∓ iy₀) of the DH Ξ-analogue lands on the real axis at some t\. Every measured t\ is a lower bound for the DH analogue of the de Bruijn–Newman constant, Λ_DH := inf{t : H_t^{DH} real-rooted}. Rodgers–Tao proved Λ_ζ ≥ 0 ("RH, if true, is barely true"); the flow-repair times measured here put a number on the other side of the same coin: DH fails RH, and in flow time it fails it barely.

The construction (derived in the probe, not recalled)

F(s) = (π/5)^{−(s+1)/2} Γ((s+1)/2) f(s) is entire with F(s) = F(1−s) and no pole terms (unlike ζ — the counterexample is cleaner here). The Mellin split of Σ n aₙ e^{−πn²x/5} at x = 1 should give

Ξ_DH(z) := F(1/2 + iz) = ∫₀^∞ Φ_DH(u) cos(zu) du, Φ_DH(u) = 4 e^{3u/2} Σ_{n≥1} n aₙ exp(−π n² e^{2u} / 5),

with Φ_DH even exactly when ω(1/x) = x^{3/2}ω(x), i.e. exactly when the functional equation holds. The probe does not trust this derivation: the constants (c, a) in H₀^{DH}(z) = c·Ξ_DH(a·z) are measured from the data, H0_vs_Xi-style, and the evenness of Φ_DH is measured as a defect of the raw unfolded series. Off-line zeros map to z = γ − i(β−1/2): complex zeros of H₀^{DH} at distance y₀ = β − 1/2 from the axis.

Instruments

probe.py → results.json + flow_repair.png. Vocabulary: measured, observed, decided; the reserved enclosure word appears nowhere in this directory (everything here is the accurate/float-of-mpmath regime — this hunt makes no enclosure claims at all).

1. Route agreement at t = 0. H^{DH}(z, 0) by quadrature vs zeta.epstein.completed_dh(1/2 + iz) by Hurwitz zeta — two code paths sharing nothing. Constants (c, a) fitted and snapped to rationals.
2. Contour moments through the collision. For a circle around a pair, p_k = (1/2πi)∮ z^k H'/H dz gives N = p₀ (must be an integer, must be 2), e₁ = p₁, and the discriminant Δ = 2p₂ − p₁². For a conjugate pair Δ = −4y² < 0; after landing, Δ = gap² > 0; Δ(t) is analytic through the collision even though the individual zeros have a square-root branch point there. t\* = the root of Δ(t). No root-chasing near a double zero, ever.
3. The survey. t\* for the off-line quadruples the literature names (Spira 1994: heights 85.70, 114.16, 166.48, 176.70; Balanzario– Sánchez-Ortiz 2007: 240.40, 320.88, 331.05, 366.64, 411.80), each re-polished in-tree by mp.findroot on dh_f from the literature seed before use, with a winding-number identity check.
4. Null control (the surrogate). The N-body zero dynamics ż_k = 2Σ_{j≠k} 1/(z_k − z_j) — no arithmetic in it, just the measured t = 0 zero configuration — integrated in the collision-safe variable Q = Δ_pair/4 (dQ/dt = 2 − 4Q Σ_w 1/((x−w)² − Q), analytic through Q = 0). If this predicts the PDE's t\* to ~1%, the repair time is generic zero-geometry, not arithmetic.
5. Rival = instrument validation on ζ. The same generic-Φ evaluator fed zeta.heatflow.Phi must reproduce zeta.heatflow.H_t to quadrature accuracy, and the moment machinery around two real ζ-flow zeros must return integer N = 2 and Δ > 0 matching zeros_of_H_t positions.
6. Lesions. (a) A contour deliberately clipping one pair member: the integer-winding check must refuse, loudly. (b) After landing, the two newborn real zeros form a close pair invisible to a default mean-spacing/20 sign scan — measure how long past t\* the standard instrument stays blind (hunt #3's blind spot, now with a clock on it).
7. Precision response. t\* re-measured at three working precisions and two contour node counts; Δ(0) against −4y₀² from the 50-digit pinned zero. A real quantity pins; an artifact wanders.

Pre-registered predictions (written before any flow was run)

From the pinned/literature zeros, y₀ = β − 1/2, naive isolated-pair model t\*₀ = y₀²/2 (from ẏ = −1/y, the mirror's pull alone):

Predictions this hunt can lose on:

• P1. Every measured t\* lands strictly below its t\*₀: every zeta-like zero of the configuration (mirror, real neighbours, mirror quadruple) pushes the pair toward the axis, so crowding only accelerates. Expected shave: ~5–15%, growing with height as the real-zero density log-grows.
• P2. dΔ/dt at t = 0 is ≈ 8 for an isolated pair; measured slope is 8 + 16y₀²Σ_w 1/((x−w)²+y₀²) + (mirror terms) ≈ 9–10 for pair 1.
• P3. The ODE null control reproduces each PDE t\* to ~1%: the repair clock reads zero geometry, not arithmetic.
• P4. The survey's max — the measured lower bound for Λ_DH — comes from pair 5 (height 240.4), not the famous pair 1, at t\* ≈ 0.06 ± 0.005.
• P5. Λ_DH ≥ max t\* ≈ 0.06, which sits inside [0, 0.2] — the interval that bounded Λ_ζ before and after Rodgers–Tao. In flow time, the counterexample is closer to satisfying RH than ζ was known to be to either side of it for most of a century. Gate-#3 moral, quantified: "Λ is small" is not a property that separates ζ from a function where RH is false.

Rules of engagement

Repo-wide rules (.venv python, mp.workdps, Agg before pyplot, honest scope). Λ_DH here is a defined quantity for a rival function; measuring it neither supports nor threatens RH (Littlewood, docs/08), and a measured collision time is a lower bound for a sup over infinitely many quadruples — the probe measures nine and claims nothing about the tail. All numbers are the accurate regime: mpmath floats with measured cross-route defects, no enclosures. If a computation here appears to settle anything open, the correct inference is a bug.