Counting What a Wallbreaker Had to Rule Out: Hidden-Plan Spaces and the Half-Life of Concealment in the Four Wallfacer Files

1. Introduction

The memoised recursion approach to counting the parses of a staged grammar took its current form with Mbeki and Ardizzone (7 BkE), whose method requires only a transition rule over the positions of a record. We apply it to the observable conduct of the four Wallfacers, i.e. to the question of how many distinct hidden plans each record of conduct left open to a Wallbreaker, and how quickly the evidence that a Wallbreaker gathered closed them.

We write from the Bunker Era, Year 14, when a photoid strike has destroyed Trisolaris, the cities lie in the shadow of the gas giants, the Wallfacer Project is long concluded, and the Luo-Ji / Ye framework has a live rival in the hypothesis of the Lurker (also 'Spirit', 'Spirit Mother' in some translations). Each of the four Wallfacers held a plan that was concealed even from the UN Planetary Defense Council (PDC), and each was assigned an ETO Wallbreaker whose task was to expose it. The quantum-phantom strike of Frederick Tyler and the stellar-bomb hostage plan of Manuel Rey Díaz were exposed, after which Tyler took his own life by drowning and Rey Díaz was stoned to death. Bill Hines's mental seal was exposed by his wife and Wallbreaker, Keiko Yamasuki, who showed that it was meant to instil defeatism and not conviction of victory. No Wallbreaker exposed Luo Ji's plan, and it became the Dark Forest Deterrence.

The four cases have been told one at a time (Pelletier, 12 BkE), and the literature on signalling has treated concealed purpose as a contest between two parties (Varga-Noor, 10 BkE), while nobody has counted the purposes that a record of conduct left open. This is the human version of the problem that we treated for civilisations under the chain of suspicion (Teillard-Osei, 9 BkE), with the hidden object now a person's purpose, and the grammar of our earlier work (Teillard-Osei, 4 BkE) carries over with new stage sets.

Three quantities are reported for each Wallfacer, namely the size of the plan space before evidence, its size after the evidence that the files record, and the number of truthful items that would halve its logarithm, which we offer as a measure of concealment depth.

2. Methods

Given a set of eight goal classes and a record of n coded actions, a plan is a goal class together with a role for each action, and the role is either decoy or a stage of the class. The action alphabet has seven codes, namely requisition (R), recruitment (H), construction (K), experiment (E), travel (T), public statement (P), and routine conduct (L). Each goal class is a chain of two to four stages and each stage admits a set of codes, as Table 1 shows. The classes are the strike capability of the quantum-phantom plan (QS), the stellar-bomb hostage plan (SB), a mental seal toward conviction of victory (MC), a mental seal toward defeatism (MD), deterrence by threatened disclosure (DT), an escape or shelter project (ES), a concealed signalling project (SG), and a programme of disinformation alone (DS). The stage sets are our own coding, and the classes MC and MD share one chain by definition.

The codes R, H, T, P, and L are feignable, meaning that a Wallfacer can perform them at small cost without pursuing their usual purpose, while K and E are not. A feignable action may be a decoy at any point. Any other action must belong to the stage set of its stage, stages are visited in order, an advance may pass over one unobserved stage, and at least one action must be assigned to a stage. We count the labellings of a record of n actions by recursion over positions, with the current stage as the state, in exact integer arithmetic. Distinct labellings are distinct plans, so the count N is the number of hidden plans that the record admits.

Evidence comes in three kinds. A role certification fixes one action as instrumental (I) or as a feint (V), a goal exclusion (X) removes one class, and a disclosure (Z) fixes the class and so codes exposure. An item that arrives after action p is applied to the prefix of p actions. We report the size of the space in bits, log₂ N, and the concealment index κ, the ratio of log₂ N to log₂ F(n), where F(n) is a type-blind ceiling in which every code is admitted at every stage and every action may be a decoy. The plans admissible conditional to the evidence received are those that agree with every item. The retained fraction ρ is log₂ N after evidence over log₂ N before it, on the same prefix. A truthful schedule uses only items that agree with the archival role labelling of the record, and the half-life H is the number of items of a best-first truthful schedule that brings log₂ N to half of its value before evidence. Best-first means that each step takes the item that lowers N most, with ties broken by position, so that H is an upper bound on the shortest truthful schedule.

The records are our coding of the action logs of the four files in the Wallfacer Program Archives, declassified series (Wallfacer Program Archives, Records Office, 33 DE), with the order of actions as logged. The evidence items are the dated observations that the files list, counted without attribution to any person. For Luo Ji we make no statement about who gathered them, and we name no Wallbreaker for him.

Three acts of Luo Ji's Wallfacer conduct are coded twice. In the main coding the broadcast of the coordinates of star 187J3X1 (the spell) is a public statement, the long wait for that star's destruction is routine conduct, and the deterrence prototype is the last of his three recorded requisitions (our identification). In the recoding the broadcast is an experiment, the wait is two experiments, and the prototype is a construction. The nuclear devices placed in solar orbit belong to his Wallfacer-era conduct as well, and we make no statement about whose authority asked for them.

Some properties of the results are fixed by the grammar for every record, and others are decided by the coding. We say in Section 3 where each property falls, and we do not count the first kind as findings.

3. Results

Table 2 gives the coded records. Tyler's file holds 36 actions, 24 of them constructions or experiments, Rey Díaz's holds 38 with 17, and Hines's holds 30 with 12 experiments and no constructions. Luo Ji's holds 42 actions and, in the main coding, none is a construction or an experiment; 31 are public statements or routine conduct, 7 are travel, 3 are requisitions, and 1 is a recruitment. The feignable share is therefore 0.33, 0.55, 0.60, and 1.00 for the four files in turn, and the records admit 3, 3, 6, and 8 of the eight goal classes.

Sizes are in Table 3. Before evidence the plan spaces hold 9,060, 96,432, 1,858,242, and 32,091,138,372,414 plans. The records differ in length, so we compare them through κ, with ceilings of 47.9, 50.1, 41.3, and 54.5 bits, and it takes the values 0.27, 0.33, 0.50, and 0.82. The step from Hines to Luo Ji is a dramatic increase, and it is larger than the whole span from Tyler to Hines.

The archived evidence is small, with 9 items for Tyler (8 role certifications and 1 exclusion), 10 for Rey Díaz (9 and 1), 7 for Hines (5 and 2), and 9 for Luo Ji (7 and 2). After them the spaces hold 196, 1,010, 89,856, and 1,811,990,528 plans, with κ = 0.16, 0.20, 0.40, and 0.56 and retained fractions of 0.58, 0.60, 0.79, and 0.69. Luo Ji's space lost the most bits, 14.1 against 4.4 to 6.6 for the others, and it ended about 9.2 million times larger than Tyler's, 1.8 million times larger than Rey Díaz's, and about 20,000 times larger than Hines's.

Under truthful best-first evidence the half-life is 6 items for Tyler, 4 for Rey Díaz, 10 for Hines, and 13 for Luo Ji, and 2, 2, 10, and 25 items bring the spaces below 2¹⁰ = 1,024 plans. The archived supply of 9, 10, 7, and 9 items exceeded the half-life for Tyler and Rey Díaz and fell short of it for Hines and Luo Ji. None of the four archived schedules reached its half-life, since every retained fraction is at least 0.58.

A common budget of nine truthful items, the archived supply of Tyler and Luo Ji, spent best-first would have left 3.0, 3.2, 10.5, and 25.6 bits. The gap between Luo Ji and Tyler after the archived evidence is 23.1 bits, and 22.6 bits of it, i.e. 98%, remain at the common budget; the shares against Rey Díaz and Hines are 108% (20.8 bits of gap, 22.4 at the common budget) and 106% (14.3 and 15.1); a share above 100% means that equal evidence would have widened the gap, which is the opposite of what a thinner supply of evidence for Luo Ji would predict. Record length does not drive the comparison, since in κ the gap to Tyler is 0.41 both after the archived evidence (0.56 against 0.16) and at the common budget (0.47 against 0.06).

Coding exposure as a disclosure, the count falls from 196 to 196 plans for Tyler, from 1,010 to 530 for Rey Díaz (0.9 bits), and from 89,856 to 44,928 for Hines (1.0 bit). The archived evidence had already left no plan in any other class for Tyler, only QS and SB open for Rey Díaz, and only MC and MD open for Hines. The Hines disclosure is therefore worth exactly one bit, the equality of MC and MD at work.

An interesting observation is that no role certification could have supplied that bit, and that his Wallbreaker supplied it from her knowledge of him as his wife.

Three of these results are structural and the record decides none of them. First, MC and MD share one chain, so their counts are equal for every record and every set of role certifications, and only a goal exclusion or a disclosure can separate them. Next, κ cannot exceed 1 because F(n) bounds N from above. Finally, a certification of a construction or an experiment changes no count, since those codes cannot be feints, so role certifications can act only on the feignable share of a record.

The remainder is decided by the coding, and Table 4 recomputes κ before evidence under five other rules. If L is costly the order of the four files is unchanged. If only T, P, and L can be feigned, the records of Tyler and Rey Díaz admit no plan at all, and the rule is rejected by them. If E can be feigned, Tyler rises to 0.60 and passes Rey Díaz at 0.38, while Hines (0.80) approaches Luo Ji (0.82); that row cannot move Luo Ji, whose main coding holds no E.

Recoding Luo Ji's three acts is the test that bears on him. His feignable share falls to 0.90 (38 of 42 actions) and his record admits 4 classes, DT not among them, because our stage sets for DT contain no experiment or construction. His κ is then 0.39, below Hines at 0.50 and above Tyler and Rey Díaz. If DT is allowed an experiment at its second stage and a construction at its third, the record admits 5 classes and κ is 0.62, the highest of the four. We report the recoding before evidence only, since the archived feint certification of the 18th action cannot stand if that action is an experiment.

4. Discussion

Luo Ji's plan space stayed large most likely because his recorded conduct was of a kind that every goal class could absorb, and probably not because less evidence reached it. In the main coding all of his 42 actions were feignable, the record admitted all eight classes, and the same number of truthful items would have left it at 25.6 bits against 3.0 for Tyler; the recoding of three acts leaves 38 feignable and four classes, and the explanation then rests on the stage sets for DT. The collapse of the Tyler and Rey Díaz files was in the same sense a product of what their records contained, since constructions and experiments cannot be feints and each one confines the plan to the few classes that use it.

The plausible alternative is that the files simply held less evidence about Luo Ji. We find that less likely, because 98% of the gap to Tyler remains at the common budget, and because his supply of 9 items equals Tyler's. A second alternative is that we coded the records with the outcomes in view, which could have pushed Luo Ji's conduct toward ambiguous codes; he stays above Tyler and Rey Díaz under every rule in Table 4 (where they admit any plan), but his place above Hines fails under the recoding unless DT admits an experiment and a construction, and the order of the two smallest spaces fails under the E rule, so we cannot rule the second alternative out.

An objection from the structure of the grammar is that a record without constructions or experiments is bound to admit more classes, so that the ordering is built into the stage sets. We concede that part of it is, as the structural results of Section 3 state, and the E row of Table 4 shows how far the order moves when that is relaxed, with Tyler more than doubling. What the stage sets do not fix is how much of a record's size evidence can remove, and the comparison at a common budget, which holds the evidence fixed, still leaves 98% of the gap to Tyler within the main coding.

The Hines file qualifies the use of size as a predictor of exposure. His space after evidence was the second largest, at 16.5 bits, and it was nonetheless exposed, because the one bit that role certifications could not supply came from his Wallbreaker's knowledge of him as his wife, a source that lies outside a measure built on conduct. As ecumenicals we take the two frameworks to be compatible, and we place cosmic sociology inside a Lurker-shaped universe as the machinery by which persons and polities hide from one another; a count of human plans says nothing about a cosmogonic agent, and we claim nothing about one here (on cosmogonic residue in counts, see Taverner, 6 BkE).

What the measure shows is narrower than a theory of concealment. The plan space of a person who acts in ways that every purpose can absorb is large and shrinks slowly, and one whose work requires buildings and trials is small from the start.

5. Limitations

The counts describe our model and four coded logs, and they carry no claim about the Project as a whole.

Our goal set of eight classes and its stage sets are our own coding, and the class DT is present because the outcome is known at this vantage, so that a Wallbreaker's own space of hypotheses was probably smaller and differently shaped than the one we count. One reader who knew how each case ended coded the records, including the recoding of Luo Ji's three acts. The archived items are counted as the files list them and not as a Wallbreaker met them, so that their timing is an ordering by action and not a date.

Where the coding ends, the reach of the method ends. First, a second reader should code the logs independently, so that the feignable shares and admitted classes in Table 2 can be compared across coders. Next, the goal set should be fixed before the outcomes are consulted, for instance from the hypotheses that each Wallbreaker's file recorded. Finally, exposure by a Wallbreaker with private knowledge of the Wallfacer should enter the model as an event with its own probability, because the Hines file shows that a large plan space and an exposed plan can coexist.