Run of 2026-08-07, probe.py, backends ['mpmath.iv', 'python-flint'], raw numbers in results.json. Probe language throughout: measured, observed, decided (= an enclosure returned lo > 0 or hi < 0).
1. The near-miss is real, decided on both backends
At the exact rationals 7005.02, 7005.0819, 7005.15 the sign of Z was decided at 64 bits as −, +, − by both backends independently. With continuity of Z that brackets ≥ 2 critical-line zeros inside an interval of width 0.13 — the pair — and the bump's clearance is decided positive: both backends agree
Z(7005.0819) = 0.003967335016595021… (midpoints agree to all 16 digits)
The dense window scan (81 samples on [7004.9, 7005.3], step 0.005) found exactly 2 sign changes, zero undecided samples, on both backends. The window contains no other zeros (γ₆₇₀₈ = 7004.04, γ₆₇₁₁ = 7006.74), so the lower bound 2 meets the strip count from the float regime.
Backend cost asymmetry, measured: the same 81-point scan took 0.028 s (python-flint) vs 28.7 s (mpmath.iv) — the ~1000× gap AGENTS.md warns about, observed directly.
2. Lesion: the default grid policy is blind to the pair 1 time in 5
The default scan step at this height, mean_spacing/20 ≈ 0.0448, is wider than the whole Lehmer gap (0.0377). Sweeping the window phase through five offsets spanning one step: 4 phases reported 2 changes, 1 phase (shift 0.027) reported 0 — every sample decided, and the count is an honest lower bound both times, but at that phase the instrument simply cannot see the closest pair below 10⁴. A sign-change scan misses, never invents; here is the miss, measured. Anyone running a default-step scan across t ≈ 7005 should know the pair is invisible at some phases.
3. Precision response: the standing rule, passed
Enclosure width at the bump vs prec_bits (python-flint / mpmath.iv):
| bits | flint width | iv width |
|---|---|---|
| 32 | 1.7e-2 (straddles 0) | 6.3e-3 (decides!) |
| 64 | 4.1e-13 | 1.5e-12 |
| 128 | 2.2e-32 | 7.9e-32 |
| 256 | 6.7e-71 | 2.3e-70 |
Width shrinks like ~2^−prec while the midpoint stays pinned — a real quantity responding to precision, per the ROADMAP standing rule. Two details worth keeping:
- At 32 bits flint straddles zero but mpmath.iv decides — the iv enclosure happens to be tighter at very low precision. The backends disagree about decidability, never about the value; that is the designed failure mode (return 0, escalate), observed at its boundary.
- Sliding the probe point toward γ₆₇₀₉: at distance 10⁻³ … 10⁻⁹ both backends decide at 64 bits. At 10⁻¹², where |Z| ≈ 5×10⁻¹³, flint still decides at 64 bits (width 4.0×10⁻¹³ — a margin of ~20%), iv needs 128. The decision cost grows exactly when it must and not before.
4. Rival (gate #1): the counterexample kills the magnitude heuristic
The prediction going in was that Davenport–Heilbronn's off-line zero at 0.8085 + 85.6993i would show a "failed Lehmer bump" — Z_dh sneaking close to zero and not quite crossing. Measured: the closest approach on [85.2, 86.2] is −0.3566 at t = 85.707. Not close at all — two strip zeros hide behind an extremum that misses zero by two orders of magnitude more than ζ's bump clears it (+0.004).
So the hunt's most useful observation is negative, and it is the rival that provides it: "|Z| gets suspiciously small" is not the signature of zeros near or off the line — crossing is binary and magnitude is a distraction. A claim of the form "near-misses of Z flag danger" would pass ζ at 7005 and fail to flag DH at 85.7, which is precisely the kind of claim gate #1 exists to kill. The instruments that survive this contrast are the ones the repo already trusts: sign counting on the line against strip counting (argument principle), never |Z| thresholds.
Standing-checklist accounting
- Rival: run, and it produced the headline (§4).
- Decoy/surrogate: not applicable — no arithmetic-vs-null claim is being made; the probe measures instrument behaviour at named points.
- Lesion: run (§2); the planted challenge is the pair itself and the default grid fails it at 1 of 5 phases.
- Precision response: run (§3), passed.
Disposition
Instrument (probe.py, scan_signs) retained; no claim promoted; no ledger entry (nothing here is a conjecture — it is a portrait of the closest call, drawn with error bars). Nothing in this note is evidence for RH (Littlewood, docs/08). Candidate for the spine, if anyone wants it: the default-step blind spot (§2) is an honest sharp edge of the packaged grid scanner worth a line in its docstring — that change belongs to zeta/, not to this hunt.