A Passive Archival Search for Curvature-Drive Trails in Pulsar Timing: Detection Limits, Recovery Efficiency, and the Candidates That Remain Unexcluded

1. Introduction

We study the archived timing record of the Solar System using the matched filter for transient delays in pulsar arrival times, which was set out in the Deterrence Era as a means of separating clock faults from real events at timing stations (Tavernier, 52 DE). We apply it to the search for the residual trail of a curvature-drive vessel. The paper is written in Bunker Era, Year 14 (BkE 14), with every record in it closed by BkE 13.

A trail is a region of space along a vessel's path in which the speed of light has been lowered and stays lowered after the vessel has gone. A signal that crosses such a region is slowed in proportion to the length of the chord it traverses, a delay which behaves as an achromatic refractive effect (Okwuosa, 12 BkE). If the limiting speed in the trail is c divided by one plus δ, a chord of length d delays the signal by dδ/c. The most extreme lowering treated in the literature is that of a black domain, a limiting speed below 16.7 km/s, which corresponds to δ of at least 1.8 × 10⁴ (Rauch-Ibáñez, 11 BkE).

Curvature-drive research has been restricted since the opening of the Bunker Era, because the trail is the feature that exposes a civilisation to any observer who looks for it. The restriction was set out in an advisory of the UN Planetary Defense Council (PDC), carried by its Secretariat, whose text we use (UN Planetary Defense Council, 3 BkE). We therefore have no controlled emission to calibrate against, and we neither propose one nor use any restricted record. The search is passive, and it asks only what the pulsar timing archive, collected for other purposes over several decades, could have shown and did show.

We define the trail model and the statistic, report what the archive contains, give the injection recovery that converts a null result into a bound, and list the candidates that the data cannot exclude.

2. Methods

Our data are the arrival-time release of the Yunnan Astronomical Observatory timing group (Yunnan Astronomical Observatory Archive, 14 BkE), which holds 38 millisecond pulsars observed from Deterrence Era, Year 20 to Bunker Era, Year 13 (DE 20 to BkE 13) in 9,731 scheduled sessions. Of these sessions 9,412 produced a usable arrival time and 319 produced none, and we call the 9,412 the usable set. Fourteen bright pulsars supply 5,288 of them and 24 faint pulsars supply 4,124. The median single-arrival uncertainty is 1.5 µs, 34% of arrival times have an uncertainty of 1 µs or better, and the mean integration of 1,950 s makes the usable set about 212 days of on-source time. The pulses have widths of a few hundred microseconds. Residuals were taken from the timing-model fits that the group published, which absorb slow red noise (Pellegrin, 33 CE). Each session record carries the station, the observing band or bands, the integration time and a free-text cause for any loss, so that every filter below could be applied without recourse to any restricted holding. The archive is not a survey designed for this question, and its sampling follows the needs of the timing programme, with bright pulsars revisited far more often than faint ones.

The record spans the period of sophon activity. Sessions before the Broadcast Era were recorded while sophons were active, and in the Deterrence years they were partially suspended by Trisolaris under the Dark Forest Deterrence. In the brief Post-Deterrence Era sophons were unconstrained, and the station, unstaffed during the Australian Resettlement, logged no sessions. A sophon could have suppressed or forged a timing excursion at any epoch, including the Broadcast and Bunker Eras, and nothing in a single-station record can exclude this, since a sophon that knew our filter could keep a forged excursion below its threshold. Every bound we report therefore holds only against an adversary that did not tamper with the record. We therefore report the bound for the full set and for the Broadcast- and Bunker-Era subset separately. The subset holds 3,138 of the 9,412 usable arrival times, 1,763 from bright pulsars and 1,375 from faint ones, and we justify it only as the part of the record collected after the Trisolaran occupation ended, with no claim about sophons.

A simple trail model, which is our own, fixes the geometry. The trail is at rest in the frame of the Solar System and has a uniform fractional lowering δ across its section. Its representative chord is d = 40,000 km, corresponding to a trail radius of 20,000 km, with a thin case of d = 4,000 km treated alongside it. The line of sight to a pulsar moves through the trail at the orbital speed of the Earth, 29.8 km/s, so that the crossing lasts τ = d/v, or 1,342 s for the representative chord and 134 s for the thin one.

Because an arrival time integrates over a window of length T, a crossing shorter than the window displaces only part of the pulses, and the recorded delay is diluted by the factor min(1, τ/T) before it is compared with the uncertainty of the arrival time concerned. The dilution is linear only while the delay is small compared with the pulse width, and for larger delays and τ/T well below one the fit follows the undelayed pulses. For the representative chord the crossing is shorter than the window in most sessions, so the filter, which is a boxcar of length τ, collapses to a single-arrival statistic. Longer crossings would need adjacent arrival times, and none of the cases we treat does. The bounds we quote therefore hold for a chord of 40,000 km and 10^-3 ≤ δ ≲ 10^-2, above which the delay approaches the pulse period and, at black-domain depth, exceeds the integration.

The statistic is the residual divided by its published uncertainty, z. A trail can only delay an arrival, so only a positive z can indicate one, but we kept the two-sided Bonferroni threshold for a family-wise level of 0.05 over 9,412 arrival times, which is z = 4.55 and conservative for a one-sided signal. Values with |z| between 3.5 and 4.55 were kept as sub-threshold candidates and triaged by three tests. A negative residual is excluded first, since a trail cannot advance an arrival. A candidate with concurrent data from a second station was checked for an excursion there, since a trail wider than the baseline between stations would appear at both. A candidate with data in two observing bands was checked for chromatic behaviour, which dispersion by the interstellar medium would show and a trail would not, and because a clock fault would pass this test we count it as informative only against dispersion.

Recovery efficiency ε was computed, and not simulated, as the mean over arrival times of Φ(g·dδ/(cσ) − 4.55), where g is the dilution factor, σ the uncertainty, and Φ the standard normal distribution function. We computed it for the full set and for the subset. Given ε and N arrival times, a zero count gives a 95% upper bound of 2.996 on the expected number of crossings, so that the bound on the fraction of observations which crossed a trail is 2.996/(Nε). If both unexcluded candidates were real the numerator becomes 6.296. That last case uses the efficiency at the detection threshold of 4.55, although the candidates were selected at 3.5, and so it overstates the bound.

3. Results

The search found no trail, and the largest excursion, z = −3.96, has a Šidák-corrected probability of 0.51 for its magnitude, or 0.71 by Bonferroni. No candidate reached the threshold.

Residuals matched what noise predicts across the whole distribution. Twenty-eight arrival times had |z| above 3, against 25.4 expected from a Gaussian, with a Poisson probability of 0.33 for a count that high or higher, and five had |z| above 3.5, against 4.4 expected, with a probability of 0.44. Because a trail only delays, the relevant count is the positive one, and three of the five excursions were positive, against 2.2 expected for z above +3.5, with a Poisson probability of 0.37 for three or more. The largest positive excursion, z = +3.74, has a one-sided Šidák-corrected probability of 0.58.

Triage of the five sub-threshold excursions left two unexcluded. The two negative excursions are excluded at once, since a trail cannot advance an arrival. Of the three positive ones, one had concurrent data from a second station, where its residual was |z| = 0.6, so that it was local to the first station and is excluded as a trail. Of the remaining two, one was observed in a single band and offers no discriminating test. The other was observed in two bands with a delay that was achromatic within the errors, which rules out dispersion and leaves a clock fault and a trail equally possible.

Sessions that produced no arrival time form a second channel. The observatory logs assign an instrumental cause to 296 of the 319 lost sessions and none to the remaining 23. A trail of black-domain depth across the representative chord would delay a signal by about 2,400 s, longer than the mean integration, and would present as a lost session and not as a late arrival. We cannot say what fraction of deep crossings would be logged as losses, so the 23 are listed here and not weighed.

Recovery efficiency and the resulting bounds appear in Table 1, and the efficiency follows in a natural way from the dilution and the uncertainty of each arrival time. For the representative chord efficiency is 0.57 at δ = 10^-4 and 0.96 at 10^-3. It reaches 1.00 at 10^-2, the upper edge of the window, where the delay of 1,334 µs approaches the period of a millisecond pulsar. Deeper trails leave the regime of the filter, and at black-domain depth the delay exceeds the integration and appears in the lost-session channel above. At δ = 10^-5 the nominal delay of 1.3 µs lies below the median uncertainty and efficiency is 0.007, so the search says nothing about so shallow a trail.

For the thin chord the same efficiencies need a hundredfold deeper lowering, because the crossing is short and the dilution is severe. At those depths the delay is no longer small against the pulse width, the linear dilution regime is not satisfied, and the two thin-chord rows at 10^-2 and 10^-1 give upper bounds on ε and lower limits on the bound.

The Broadcast- and Bunker-Era subset has an efficiency of 0.96 at δ = 10^-3, as for the full set, and a zero-crossing bound of 9.9 per 10,000. That is the figure to quote if the earlier sessions are set aside.

4. Discussion

Earlier work derived the delay that a trail imposes and noted that pulsar timing could in principle register it (Okwuosa, 12 BkE). It could not say how much of the record had been exposed to the test, because it had no account of dilution by the integration window and no recovery calculation. Thanks to the efficiency computed here, the null result becomes a statement about a fraction of observations and not only about a count of detections. For a chord of 40,000 km and a lowering from 10^-3 to 10^-2, no more than 3.3 in 10,000 of the observations crossed a trail, at 95% confidence, and the bound for the Broadcast- and Bunker-Era subset is 9.9 in 10,000.

That statement is narrow. It holds for the trail model of Section 2, for lowering inside the window stated there, and for the pulsars and epochs in the archive. It does not bound the number of vessels that have used a curvature drive, because that would require a trail lifetime and a sky geometry which we have not modelled. Nor does it exclude a sophon that suppressed or forged an excursion at any epoch, including the Broadcast- and Bunker-Era subset, and it holds only against an adversary that did not tamper with the record.

A school position bears on how a null result is read, and ours is declared here. We write within the Lurker framework, in which the persistence of a drive trail is a constraint fixed at the cosmogonic scale (Sakamura, 12 BkE), and in this paper the Luo-Ji / Ye axioms are treated as local-domain-only. The search does not depend on that position, and an author who works within the Luo-Ji / Ye framework would compute the same bounds from the same archive. The position bears instead on policy, since a trail that persists cannot be recalled by its maker, and the early Bunker-Era restrictions look well founded to us for that reason. The present search adds nothing to that judgement, and the restrictions would stand on the same footing had the archive shown a trail.

Two unexcluded candidates are not evidence. The three positive excursions are what noise predicts, the triage could not separate the remaining two from clock faults, and the data contain no feature that would favour a trail over an instrumental explanation. Neither can the 23 unexplained lost sessions be weighed, since we hold no baseline rate for them. No trail crossed the 38 sampled lines of sight within the reach of the filter.

5. Limitations

Only trails whose lowering and chord produce a diluted delay of several microseconds, and no more than a fraction of the pulse width, can be seen by this search. Shallow lowering, at 10^-5 or below, is invisible to the instruments. Thin chords would need a lowering a hundred times deeper, outside the regime in which our dilution model applies, and these limits come from the timing precision of the archived receivers and not from the method.

Our trail model is a representative case and nothing more. We assumed a uniform section, a trail at rest in the frame of the Solar System, and an Earth-orbital crossing speed, and a trail that drifts, that has structure along its section, or that crosses the line of sight at a small angle would change the efficiency. A crossing that straddles the edge of a window would split the dilution factor g between two windows, which our single-window treatment does not allow for. We did not model these cases.

Second-station data existed for one of the three positive candidates, which left two with no clean test, and a clock fault and a trail cannot be separated by any test made within a single station. The same record cannot exclude a sophon acting at any epoch, including the Broadcast- and Bunker-Era subset. The dropout channel is uncalibrated, and the archive covers 38 pulsars, so that the lines of sight sampled are a small and fixed set on the sky.

The present search is one step toward a bound on trail traffic near the Solar System, and the next calculation is the delay of a trail that moves relative to it, for which the crossing time and the dilution both change and the present efficiency cannot yet bound.