Redundancy Planning for the Gravitational-Wave Transmitter Network: Placement, Hardness and Coverage Under a Modelled First Strike

1. Introduction

In the model of this paper, a network of six regions delivers the required broadcast energy within 12 h with probability 0.978 when three targets are attacked before the broadcast begins, at a cost of 43 units. A network of four regions, built at 41, delivers it with probability 0.238. Most of the difference is set by where the transmitters stand, and a much smaller part is played by the hardness of the individual sites. The best network of 12 transmitters in eight regions reaches 0.999.

The Dark Forest Deterrence is only as good as the broadcast that it threatens. The threat is to send the position of Trisolaris through gravitational-wave transmitters, and the dead-man switch that releases the broadcast has been held by the Swordholder since the establishment of the Dark Forest Deterrence. The network behind the switch is land-based and shipboard, and the warship Gravity carries one transmitter. A companion paper in this issue, by Sarkissian-Low, treats the credibility of the hand on the switch as a forecastable quantity. The present paper asks whether the switch itself would survive a first strike.

Written in the Deterrence Era, Year 61 (DE 61), this paper is based on records that all predate that year. No source available to it states the number, type or siting of the land transmitters, or the number of droplets in the Solar System. Every configuration and every strike count below is therefore a construct of the model, and none is a statement about the network as it stands.

Planning of this kind is familiar from energy-limited survey teams.

Units must return as much as they can inside a fixed budget, and the coverage must be kept through the loss of members (Vesnin-Wu, 52 DE). A buried transmitter, for example, is priced at 5 cost units and is expected to deliver 7.9 unit-hours in a 12 h window, whereas a surface transmitter is priced at 3 and is expected to deliver 11.5. The cheaper units are also the more exposed ones. A struck surface unit is lost with probability 0.95 and a struck buried unit with probability 0.60.

2. System Description

Each transmitter ramps from standby to rated output once the dead-man switch has been released. It then emits until the window closes (Varga-Reyes, 51 DE). Land sites follow the three-grade hardness classification of Ruiz-Ekwueme (53 DE). The surface grade is the cheapest and the fastest to ramp, the buried grade has a lower rated output and a longer ramp, and the deep grade is the slowest of all.

Land sites are grouped in regions. A region is an area small enough for one strike to reach every site in it and distant enough from the next region that no strike reaches both (Lindqvist-Haas, 54 DE). Eight regions are used, each hosting at most one site of each grade, so that 24 land candidates exist. The network is completed by three shipboard units as a model value, with Gravity carrying one, and each ship is a separate target.

The attacker is credited with the knowledge that the sophons supply. Sophons are ubiquitous and coordinate without measurable delay, so the position of every site and ship is assumed known at the time of the strike. The UN Planetary Defense Council (PDC) received and logged the Trisolaran declaration of a partial suspension of sophon interference (United Nations Planetary Defense Council, Secretariat, 2 DE). Humans cannot verify the declaration, and the model does not rely on it. A strike capacity A is the number of distinct targets that can be struck inside the window. It is a free parameter and not an estimate of the number of droplets.

Records come from the standby and output trial sheets of the Fleet International Operations Office (Fleet International Operations Office, 52–60 DE). They hold 144 standby trials and 51 output tests, coded by hardness grade and by land or shipboard. The sheets carry no site names, so no inference about the number or position of operational transmitters is drawn from them.

The kill probabilities in Table 1 are assumptions of the paper, and no record bears on them.

3. Analysis / Model

Energy of E = r(W − τ) unit-hours is delivered by a unit that is live at the strike. Here r is the rated output, τ the ramp time and W the window of 12 h, and nothing is delivered if τ exceeds W. A unit is live with probability a if it is not struck and a(1 − q) if it is struck, where a is the ready fraction and q the kill probability. Units are assumed to fail independently. The broadcast succeeds when at least 50 unit-hours are delivered by the live units. That figure is a normalisation in the manner of a design requirement for a deterrent signal (Haldane-Itō, 57 DE), not a physical estimate.

The distribution of delivered energy is computed exactly by enumeration over the numbers of live units of each grade (Okonjo-Feyi, 50 DE), on the unit values of Table 1 and without rounding. The A targets that minimise the probability of success are attacked, one strike to a target. The minimum is found by complete enumeration. For the full network, three strikes give 165 subsets of the 11 targets, namely eight regions and three ships, and all of them are enumerated. Each configuration is scored by its worst-case probability of success.

Selection of grades for the regions is limited by a cost budget of 45, and a region may be left empty. The labelled search space holds 1,175,475 configurations. Regions are exchangeable, however, so only 875 distinct configurations need to be scored. The best configuration at the design capacity of three strikes is given in Table 2, together with the best configurations of several restricted families.

Dispersion carries most of the result, and the comparison that the model decides is the one below the region count. At three strikes, with cost held near 41 to 43, the best six-region network succeeds with probability 0.978 and the best four-region network with 0.238.

When the strike capacity equals the number of regions, every land site is struck by construction, and the figure measures only survival of the kill draw. The three-region network is of that kind. It gives 0.006 at three strikes against 0.999 for eight regions, and that gap is structural.

Less of it is carried by hardening. Four buried sites are added to the surface-only network at an added cost of 20, and 0.023 is gained at three strikes, 0.137 at four and 0.620 at five. Deep sites do not appear in the optimum at budgets of 30 and 45, because a ramp of 6.3 h in a 12 h window leaves them 2.85 unit-hours. One does appear at a budget of 38, as a single addition. The budget ladder is gentle. As the budget rises from 24 to 30, 38 and 45, the best worst-case probability at three strikes rises from 0.975 to 0.984, 0.991 and 0.999, at realised costs of 24, 29, 38 and 44.

A rival explanation is that the gain from dispersion is a gain from numbers, since more regions mean more sites. The model rules this out. The five-region network holds 10 sites and the surface-only network holds eight, yet the surface-only network succeeds with probability 0.975 at three strikes and the five-region network with 0.840.

4. Validation Against Field Data

Two sets of records bear on the model. The ready fraction and the ramp time were obtained directly from the standby trials. The output tests give the energy delivered in a 12 h run at reduced emission, scaled to rated output before comparison, with no position data carried, and they were withheld from the estimation. Wilson intervals are used for ready fractions, while t intervals are used for ramp means and output ratios, because every group has fewer than 60 trials and the output groups fewer than 20. Ramp means are computed over the trials that reached readiness, 51, 38, 27 and 19 for the four grades.

Measured delivery lies below the prediction for every grade, by 2.4% for surface, 4.0% for buried, 5.3% for deep and 8.3% for shipboard units. Each grade's 95% interval for the ratio contains 1, but all four ratios lie below it. Pooled across grades by inverse variance, with normal limits, the ratio is 0.965 (95% CI 0.937 to 0.992), so a common shortfall of a few per cent is likely real. The worst case is the shipboard grade, with nine tests and the widest interval, 0.794 to 1.041.

Table 3 sets the trial summaries beside the predictions.

The consequence for the design is examined by rerunning the best network with every delivery scaled to its measured ratio. The probability of success falls from 0.999 to 0.997 at three strikes, from 0.965 to 0.939 at four, and from 0.804 to 0.700 at five. The surface-only network, derated in the same way, falls from 0.975 to 0.966, from 0.827 to 0.794 and from 0.184 to 0.176. The ranking of the two networks is unchanged.

Symmetry reduction was checked against full enumeration of the labelled configurations on 27 smaller instances that had three, four or five regions, three budgets each, and strike capacities of two, three and four. The optimum was identical in all 27. The reduced search visited 4.2 to 4.4 times fewer configurations at three regions and 40.6 to 43.7 times fewer at five. Full enumeration of the eight-region instance was not run. The reduced search scores 875 configurations where it would score 1,175,475.

5. Failure Modes

The kill probabilities are the weakest input. No record bears on them, and no source available to the paper records a strike on a hardened ground installation. For example, raising the buried and deep values to 0.80 and 0.40 leaves the best network at 0.998 for three strikes but lowers it to 0.916 at four and 0.577 at five. Values of 0.90 and 0.60 give 0.997, 0.877 and 0.405, and 0.95 for every grade gives 0.997, 0.854 and 0.301. No alternative set changed the optimum of the design search.

Ordering by number of regions was preserved under every alternative set as well. At three strikes the six-, four- and three-region networks of Table 2 fall to 0.959, 0.084 and 0.001 under the first set, 0.947, 0.035 and 0.000 under the second, and 0.940, 0.016 and 0.000 under the third, while the best network keeps 0.998, 0.997 and 0.997.

Independence is the second weakness. A fault common to every unit, such as a shared control fault, caps the probability of success at the probability that the fault is absent. For example, a common fault of probability 0.02 lowers the best network to 0.980 with no strike, 0.979 at three strikes and 0.945 at four. The value of 0.02 is illustrative and was not measured. Window length is the third. If the window is shortened to 6 h, the best network is left at 0.959 with no strike and 0.006 at three strikes, because 80% of the output of the whole network is then required. If it is lengthened to 24 h, the network holds 0.994 even at six strikes.

Strike capacity is bounded by assumption. Each target may be attacked once, and no credit is allowed for a second strike on the same target. The best network falls to 0.452 at six strikes. At three strikes the attacker's best choice is three regions that hold buried units, and no ship is struck.

Shipboard units cannot carry the broadcast. The three of them deliver at most 13.1 unit-hours against a requirement of 50, and less once the ready fraction of 0.864 is applied. They serve as targets that cost the attacker nothing to ignore and as spare output after a land strike, whereas a network of ships alone has a probability of success of zero at every strike capacity, including none.

6. Conclusion

This paper modelled the gravitational-wave transmitter network as a placement and coverage problem under cost, energy and time budgets. Every configuration was scored exactly against an attacker that knows every site. Against three strikes the best six-region network holds a probability of success of 0.978 and the best four-region network 0.238, the comparison that the model decides. The best network inside a budget of 45 holds 0.999, and 0.965 and 0.804 against four and five strikes. The same budget concentrated in three regions gives 0.006 at three strikes, which is structural. Measured delivery lies 2.4% to 8.3% below the model.

The ordering of designs is likely to hold, while the absolute values may not. The ordering by number of regions held under every set of kill probabilities tried, and the optimum did not change. The alternative explanation is that the ordering owes more to the independence assumption than to geography. The common-fault case cannot test that, because it enters as a constant factor and leaves the order unchanged by construction. A fault correlated inside regions would test it, and that test has not been done. The explanation is the less likely, since the regions of the model already carry the dominant correlation, a shared strike. The probabilities remain conditional on kill values that no record supports, and the strike capacities could be far from any real figure.

We write within the Luo-Ji / Ye framework, in which survival is the primary need of a civilisation, and the survival of the broadcast capability is treated here as a precondition of the deterrence that serves that need.

The method carries to any reserve that must act after the first loss, such as the stores and relay points of the survey fleets. The open methodological question is how a kill probability can be estimated at all when no strike has ever been observed. The single kill probability will be replaced by a distribution over footprints, and the independence of unit failures inside a region will be relaxed first.