Counting Contact Histories under the Chain of Suspicion: An Exact Enumeration of Admissible First-Contact Sequences for Two to Five Civilisations

1. Introduction

The memoised state-space approach to counting labelled event sequences was given its present form by Mbeki and Ardizzone (7 BkE), who needed nothing beyond a rule for the events that may follow each state. We apply it to the contact histories of small sets of civilisations, i.e. the ordered sequences of concealment (C), detection (D), strike (S), and broadcast (B) events that the two axioms of cosmic sociology and the chain-of-suspicion argument leave open on a given contact tree, and we report exact counts for between two and five civilisations.

Our vantage is the Bunker Era, Year 9. Trisolaris has been destroyed by a photoid strike, humanity shelters in the shadow of the gas giants, and both the Luo-Ji / Ye framework and the Lurker hypothesis remain live contenders. The framework credits cosmic sociology to two axioms, that survival is the primary need of a civilisation and that a civilisation grows and expands while the total matter in the universe stays constant, and it derives from them the chain of suspicion and the technology explosion. The chain of suspicion has been analysed one chain at a time (Hartnoll, 1 BkE; Halvorsen, 8 BkE), and the postures recorded in the Deterrence Era have been catalogued as declared concealments and disclosures (Fenwick, 5 BkE), while nobody has counted the histories that the argument permits.

Three counts follow. The first is the number of histories on a tree when strikes are excluded, the grammar introduced in earlier work (Teillard-Osei, 4 BkE). The second is the number when strikes are admitted. The third is the number that remains when the chain-of-suspicion argument is imposed as a restriction on the alphabet, and for each of them we ask what fraction of histories contain a strike and how the answer depends on the shape of the tree.

2. Methods

Given a set of n civilisations labelled 0 to n − 1 and a contact tree T on those labels, two civilisations can detect one another, conditional to the tree, only if an edge of T joins them. We take one tree of every shape on two to five nodes, namely the single tree on two and on three nodes, the path and the star on four nodes, and the path, the star, and a fork on five nodes, the fork being a centre joined to two leaves and to a path of length two, so that seven trees result. Every history is counted with the labels fixed, so that a history in which civilisation 1 conceals first differs from one in which civilisation 2 does.

A state records the set of living civilisations, the ordered pairs (i, j) that have been detected, the civilisations that have concealed, and those that have broadcast, and four events change it. A detection D(i, j) needs i and j alive, an edge between them, no earlier detection of j by i, and no concealment by j. A concealment C(i) needs i alive and neither concealed nor broadcast, and it bars every later detection of i. A broadcast B(i) has the same precondition and makes i detected at once by each of its living neighbours in T. A strike S(i, j) needs both civilisations alive and a detection of j by i, and it removes j from all further events. Each ordered pair is therefore detected at most once, each civilisation conceals or broadcasts at most once, and each civilisation is struck at most once.

We count three models on each tree. Model M0 omits the strike event and so reproduces the strike-free setting. Model M1 admits all four events. Model M2 adds the suspicion restriction to M1, under which a civilisation that has detected a living civilisation may no longer conceal or broadcast, a formalisation of the argument that once suspicion is engaged the only posture that resolves it is a strike. The rule is our own and deliberately strict, since dark-forest reasoning as Luo Ji set it out would also allow a detecting civilisation to stay hidden. The restriction is one of several formalisations that could be written, and it has the property that it only removes events and never adds them.

A history is maximal when no event can extend it, and we count maximal histories only, an arrangement that fixes the end of a history without any stopping rule. The choice has a consequence that we use below. In M1 and M2 a strike stays available for as long as a living civilisation has detected another living civilisation, so a maximal history that contains any detection or broadcast must also contain a strike.

We count by recursion over states. The number of maximal continuations from a state is the sum of the numbers from each state reached by one admissible event, a state with no admissible event counts one, and memoising on the full state (alive set, detected pairs, concealed set, broadcast set) means that each state is expanded once. Counts are carried as vectors indexed by the number of strikes and held as exact integers, following standard teaching practice (Fleet International Academy, San Francisco, Department of Discrete Mathematics, 3 BkE). The largest computation, the five-node path under M1, visited 339,253 states. As an independent check we enumerated explicit event sequences without memoisation, and the vectors agreed with the recursion for n = 2 and n = 3 under all three models and for the four-node path and star under M2 (6,639,382 and 11,733,282 sequences). The scripts use only the python3 standard library.

3. Results

Table 1 gives the number of maximal histories for each tree and model. With strikes excluded (M0) the counts are 56 histories for two civilisations and 8,672 for three, and they reach 4,942,776,960 on the five-node path and 6,599,415,552 on the five-node star. Admitting strikes (M1) raises the counts to 170 and 139,138 for two and three civilisations and to 4,071,552,745,552 and 7,278,139,534,056 on the five-node path and star. The suspicion restriction (M2) cuts the M1 counts to 52 and 10,772 for the two- and three-node trees and to 10,373,674,282 and 45,577,139,472 on the five-node path and star, so that M2 lies below M0 for two civilisations and above it from three civilisations onward.

The strike-free histories of M1 and M2 can be counted by hand. A maximal history without a strike cannot contain a detection or a broadcast, so it consists of the n concealments in some order, and there are exactly n! of them, namely 2, 6, 24, and 120 for n = 2, 3, 4, and 5, a count that the recursion returns for every tree and for both models. The share of histories that contain a strike is therefore 1 − n!/N, where N is the total in Table 1. It is 98.82% for M1 and 96.15% for M2 on two civilisations, 99.9957% and 99.944% on three, and above 99.9996% under both models on every tree with four or five civilisations.

Table 2 lists the strike-free shares themselves.

Growth with the number of civilisations is steep, and on the path each added civilisation multiplies the M0 total by 155, 493, and 1,156 in turn (from two to three, three to four, and four to five nodes), the M1 total by 818, 3,228, and 9,066, and the M2 total by 207, 616, and 1,562, a dramatic increase in the multiplier at every step.

The suspicion restriction removes a growing share of the histories of M1. The ratio of the M1 total to the M2 total is 3.3 for two civilisations, 12.9 for three, 67.6 and 47.3 for the four-node path and star, and 392.5, 159.7, and 291.9 for the five-node path, star, and fork. The restriction therefore shrinks the space of histories by more than two orders of magnitude at five civilisations on every tree, and because it leaves the n! strike-free histories untouched it raises the strike-free share by the same factor, from 2.95 × 10^-11 to 1.16 × 10^-8 on the five-node path, a value that remains below one in eighty million. The factor is largest on the path, smaller on the fork, and smallest on the star.

Tree shape matters little when strikes are absent and a great deal when they are present. Table 3 gives the ratio of each tree's total to that of the path with the same number of nodes. At five civilisations the star holds 1.34 times as many histories as the path under M0, 1.79 times as many under M1, and 4.39 times as many under M2, while the fork holds 1.12, 1.24, and 1.67 times as many. The distribution of the number of strikes depends on the tree as well, and under M2 the modal history on the five-node path contains three strikes and that mode holds 73.62% of the histories, whereas on the five-node star the mode is four strikes, i.e. a single survivor, with 63.43%. The share of histories in which exactly one civilisation survives is 17.46% on the five-node path under M2 against 63.43% on the star, and 30.68% against 66.09% under M1. The mean number of strikes per history under M2 is 3.09 on the path, 3.48 on the star, and 3.25 on the fork.

4. Discussion

The counts establish the following. The suspicion restriction removes most of the histories of M1, and the removal grows from a factor of 3.3 at two civilisations to a factor between 160 and 392 at five. The shape of the contact tree matters little in the strike-free model and a great deal in the models with strikes, the star-to-path ratio rising from 1.34 under M0 and 1.79 under M1 to 4.39 under M2, so that the suspicion rule amplifies the effect of shape. The strike-free histories of the full model are exactly the n! orderings of universal concealment, a structural property of the grammar, so that every other history contains a strike and the strike-free share at five civilisations is below one in eighty million under either M1 or M2. These are statements about the grammar, so they carry no sampling error and no estimate, and they give cosmic sociology its first exact measure of the space of histories that its central argument permits.

A natural objection is that counting treats every history as equally likely, including histories that no civilisation under the axioms would follow, and we accept the premise. The counts describe the size and shape of the space of admissible behaviour and not its probability, and the size is what the argument needs, because the claim that the chain of suspicion drives a civilisation to strike is a claim about what survives in that space once the restriction is applied. An interesting observation in this respect is that the restriction multiplies the strike-free share by 160 to 392 at five civilisations while the share of histories with a strike stays above 99.9996% on every tree with four or five civilisations. The restriction therefore does not create the dominance of strikes, which follows from maximality alone. What it does is remove strike-bearing histories selectively, and it leaves universal concealment hundreds of times more prominent than under M1. Under this formalisation the counts therefore lend the chain of suspicion no weight as the source of strikes. What they measure is how sharply it prunes the space and how much it magnifies the role of contact structure.

As ecumenicals we treat the two frameworks as compatible, with cosmic sociology as the local mechanism inside a Lurker-shaped universe, so that any cosmogonic agent would have to be sought in the structure of the space of histories, for example in the n! histories of universal concealment (cf. Taverner, 6 BkE), and not in the rules that generate a single history.

Tree shape matters little until strikes are admitted, and then the suspicion rule amplifies it, from a star-to-path ratio of 1.79 under M1 to 4.39 under M2. A record of who could detect whom is therefore needed before any count of strike-bearing histories can be compared across settings, although the behavioural rule moves the totals by a factor of 160 to 392 at five civilisations, against at most 4.39 for the tree.

5. Limitations

The counts hold for the model and not for the cosmos.

Maximality makes the share of histories with a strike equal to 1 − n!/N by construction, because a maximal history that leaves a detection unresolved cannot exist, and a model that let histories stop with a detection unresolved would show a lower share that our counts do not measure. Histories are labelled and ordered, every history carries the same weight, and the contact tree is known and fixed before the first event. The state count grew by a factor of 17.5 from four to five civilisations on the path under M1 (from 19,365 to 339,253 states), and we have not run larger trees.

Three limits of the method set the further work. First, the grammar fixes the contact tree in advance, whereas the set of civilisations in any real tree is itself uncertain, so the counts have to be recomputed over a distribution of trees before they are set against the historical record. Next, the histories are ordered and labelled, and the unordered classes that archival reconstructions can actually distinguish are not counted here (Fenwick, 5 BkE). Finally, the suspicion restriction is one formalisation among several, and the shares in Table 2 must be recomputed under each alternative; until that is done, the result stands or falls on the reach of that single rule.