The race to fill low Earth orbit

Six statements about the model; the mapping to the constellations is a reconstruction R.

• In a pure race, intelligence about the leader's plans buys nothing. When neither side's building affects the other's, the follower can guarantee closing the gap by $T$ exactly when its full-rate path meets the leader's full-rate path by $T$; building at the full rate is optimal for both, and the value is the same whether builds are simultaneous or one side sees the other's current build (Proposition 7.14 (p. 301)(a)). In the pursuit game of Model D2, by contrast, seeing the opponent's current move decided between capture at once and no capture at all: information about the opponent's move is valuable only when the opponent's move changes what one can achieve. In a race of this kind, a better forecast of the leader's manifest improves forecasts, not guarantees D [Proposition 7.14 (p. 301)(a), certified by backward induction under three information patterns].

• The gap is not the state. With unequal attrition, two starts with the same gap can lie on opposite sides of the bridge: from $(10,9)$ the follower closes at date $3$, from $(18,17)$ it never closes (Figure 7.5 (p. 285)a). A headline that compares counts — “the follower is so many satellites behind” — drops the levels, and with them the attrition that each side must replace before it can grow D [Proposition 7.14 (p. 301)(b)].

• Out-launching is not catching up. The base follower builds half as fast again as the leader and still has the lower ceiling, $18$ against $20$, because its units retire sooner. Its window for catching up is set by the leader's level: once the leader holds $\bar{x}=15$ units, the follower never closes from behind, at any date. Below that level it can close only early and only from close behind: from $x_0=10$ it closes from $y_0=9$, is level or ahead at dates $3$ to $10$, and is behind at every date from $11$ on; from no integer start $y_0\le 8$ does it ever close (Table 7.1 (p. 284)) D [Proposition 7.14 (p. 301)(b); “never” certified for every date].

• Build rate is the short-deadline lever, lifetime the long-deadline lever. Doubling the build rate and halving the attrition rate give the same ceiling, but the faster build gets there sooner at every date (Proposition 7.14 (p. 301)(c)(i); column $2B$ against $\mu /2$ in Table 7.1 (p. 284)). Comparing one more unit of build rate with halving attrition, the extra unit adds more up to a crossover date and the longer life adds more after it; the crossover comes earlier the larger the follower's fleet, because attrition is levied on the fleet. In the table every closing date falls before the crossover, so the extra unit of build closes first or at the same date in every row; what reverses is the capacity at a deadline past the crossover, and the closing-date ranking itself reverses only when both fleets start nearer the ceiling — from $(20,18)$ the halved attrition closes at date $2$ and the extra unit at date $3$ D [Proposition 7.14 (p. 301)(c)].

• With a second resource, no single bisector exists. When capacity requires both satellites and ground capacity, the follower's winning starts form a staircase, one step per closing date (Figure 7.5 (p. 285)b). The follower with $9$ satellite units and $\frac{23}{2}$ ground units closes at date $3$; the one with $9$ and $11$ does not close by date $8$, although by satellites alone it is past the threshold. A follower short of ground capacity must close early, before the leader grows further; a follower short of satellites can close late. The rate at which one resource substitutes for the other changes from step to step D [Proposition 7.14 (p. 301)(d)]. Spectrum priority, which the Kuiper order makes depend on the date of deployment, is a further coordinate of the same kind R.

• A milestone is a reachability question with a date, and the regulator chooses the payoff. The interim Kuiper milestone asked whether a follower could reach a target by 30 July 2026; the operator's own projection answered it before the date, and the regulator's choice was what missing it should cost: a loss of priority for the shortfall and the bond, not termination of the authorisation. In the model's terms the regulator did not move the state or the rates; it rewrote field 5, and with it the game the follower plays after the date. The ITU procedure does the same mechanically, by shrinking a filing that misses a milestone to a multiple of the number deployed R.

Capacity is not a count: satellites of different generations and sizes carry different throughput, and a count is at best a proxy for the state variable the model needs. The build rates are not exogenous bounds but the output of launch vehicles, factories and finance, each with its own strategic problem (Chapter 6's export controls are one); the model's rates are constants. Attrition is not proportional: satellites retire by cohort at the end of a design life, which makes losses depend on the launch history rather than on the current fleet, and failures cluster by batch. The players are more than two, and they are not symmetric: one leader, several followers, governments that are customers, regulators and backers at once. Above all, the comparison argument needs monotonicity (Remark 7.15 (p. 304)): a common pool of orbital shells or spectrum that the leader pre-empts keeps the leader's full rate a worst case, but building that costs money, a budget that makes early launches expensive later, or a payoff that is not the race alone, all give the leader a trade-off; then timing and information matter again and the bridge must be computed by the backward recursion, not read from two paths. And the record is about deployment, not service: the model says nothing about users, prices or coverage.