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.