Composing Learners

The individual is confined. That is not a remark about ambition; it is a proposition, and it is the reason this chapter exists. The hypothesis space is a tree with an ultrametric on it, and in an ultrametric a walk whose steps are all shorter than some radius never leaves the ball it began in. A learner that revises carefully cannot reach the theory it needs. The noughts-and-crosses measurement makes this concrete and slightly comic: the move that would rescue a confined population costs nothing and no individual can make it.

So the unit has to move, and this chapter moves it. A learner is redefined as a distribution over a population — scientists, computations holding no hypothesis at all, and reservoirs — with that population serving as a namespace. The operator that composes two such learners is parallel composition, which is to say no new machinery whatever. Everything interesting is then controlled by one dial: how much the two namespaces overlap, and in which of several typed namespaces the overlap sits.

Turned to zero, the dial gives independence, and independence turns out to be a theorem rather than a hope — though an unstable one, since two learners manufacturing names privately can collide by accident, and buying back the guarantee costs a naming discipline. Turned up, the same dial generates the standard table of ecological relations — competition, predation, mutualism, commensalism, theft — as sign conditions rather than as an inventory borrowed from biology. And it yields an asymmetry i think is the most interesting thing in the chapter: because tokens are conserved but formulae are not, cooperation can only ever be epistemic and competition is always metabolic. Two learners can both come out ahead in what they know. They cannot both come out ahead in what they hold.

23.1 Introduction

23.1.1 Why the unit moves up

The mortal scientist of \(\chSci\) is an individual: a term with a metabolic channel, a hypothesis in guard position, and a walk on the relaxation lattice. Twice in that note the individual turns out to be the wrong unit. Corollary 21.3 there observes that destructive assay leaves an individual-level hypothesis worthless after its own first test, so the subject of a hypothesis must be a namespace rather than a specimen. And Proposition 21.4 observes that an ultrametric walk of short steps never leaves the ball it began in, so the holder of a hypothesis cannot repair a shallow commitment; \S12 of that note supplies recombination, and \S17.2 states plainly that a population is required rather than preferred.

The worked example makes both visible on numbers §22.9 of \(\chGame\). The corner-opening mutant that invades the resident population differs from it by a revision of distance \(2^{-1}\), is unreachable to any individual by deepening, and is free to a lineage because the cost falls on an offspring holding no confirmations to invalidate.

So the individual is already not the unit of learning; the lineage is. This note asks what happens when the lineage is not the unit either.

23.1.2 The move, stated once

A learner is a distribution over a population: scientists, mortal computations holding no hypothesis, and resource reservoirs. That population is carried by names, and by Proposition 22.1 of \(\chGame\) those names are manufactured by quotation rather than drawn from a stock of atoms, so the family is generated by a decidable predicate and can be described. Hence a learner is a namespace, in the sense the hypothesis language already uses for the subject of a hypothesis.

Learners compose by parallel composition, because in this calculus there is nothing else for composition to be. When the namespaces are disjoint the composed learners have non-interacting capabilities. When they overlap they can interact internally, and that interaction includes both cooperation and competition.

23.1.3 What we are claiming, and what we are not

We claim that composition of learners is parallel composition on terms, union on namespaces, and undetermined on distributions; that the determination of the third is exactly the content of the interaction, and that it factors precisely when the namespaces are disjoint; that a single grade of separation is too coarse and a typed vector of grades recovers the ecological relations rather than importing them; that under conservation cooperation and competition are structurally different rather than opposite in sign; that non-interference is an observation and not a guarantee unless purchased; and that a composite’s phase is not a function of its parts’ phases, which we exhibit on exact numbers.

We do not claim to have proved a monoidal-category theorem: §23.4 states the comparison map and the conditions for it to be invertible, and leaves the coherence to the open problems. We do not claim the access-rate model of §23.10 is canonical; it is one modeling choice, flagged as such, and the alternative is worked. As in \(\chGame\), guards below are written in the ideal logic and the current interpreter accepts only the arithmetic and pattern fragment of the where clause; occurrences that specify rather than perform a check are marked.

23.2 What this chapter carries forward

The cut, the three grades of access, the conservation law, and the trophic profiles are as in the two preceding chapters and are not restated. Four things are worth having in front of the reader, because the argument turns on their exact form.

The logic is generated, and there is no restriction.

Feeding the presentation of the rho calculus to the OSLF family yields one structural connective per term former, one context-labeled modality per rewrite rule and redex position, and—because \(\mid\) carries an associative–commutative theory—a separating conjunction. A name is a quoted process, so \(n \models @\varphi \iff n = @P\) with \(P \models \varphi\): this is namespace logic [39], in which one describes a namespace by a property rather than enumerating it. Following §22.2 of \(\chGame\) we treat new as sugar and drop the hidden-name quantifier: freshness is manufactured by quotation, and graded separation is graded over a described namespace rather than over a restriction vector, \[P \models \varphi \sep{\Nsp}{k} \psi \iff P \equiv Q \mid R,\ Q \models \varphi,\ R \models \psi,\ \bigl|\, \fn(Q) \cap \fn(R) \cap \Nsp \,\bigr| = k .\] Everything below that a modal logic could not do is done with \(\sep{\Nsp}{k}\), and §23.5 turns on the choice of \(\Nsp\).

Conservation and the game.

Tokens are conserved: \(\sigma + \kappa = \sigma_0 = \Theta\), with \(\sigma\) held in live stacks and \(\kappa\) migrated into the record. There is no primary producer. Under the internalizing encoding \(\llbracket - \rrbracket : \mathcal{C}(\rt) \to \rt\) a located stack becomes a message at a metabolic channel \(m_P\), and harvest is an ordinary receipt on \(m_P\) performed by someone else; the predatory contexts are exactly the non-image contexts holding a metabolic name \(\chSci\), Theorem 21.1. A game is a choice of admitted context class \(\mathcal{A}\) with \(\llbracket \mathcal{C}(\rt)\text{-}\Ctx \rrbracket \subseteq \mathcal{A} \subseteq \rt\text{-}\Ctx\).

The hypothesis space, and confinement.

Beneath the characteristic formula \(\chi_P\) sits a lattice of progressively weaker formulae obtained by forgetting conjuncts; revision is a walk on it, the metric is the \(\nu\)-stratified ultrametric \(d(\varphi,\psi) = 2^{-n}\), and distance, modal depth, and cost are one graded structure. Confinement (Proposition 21.4 of \(\chSci\)) says a walk whose steps are all shorter than \(2^{-n}\) never leaves the ball of radius \(2^{-n}\) it began in; cost-monotonicity (Proposition 21.5) says the escaping move is the expensive one; and bounded branching (Proposition 21.6) with its corollary poverty is confining says the number of siblings a scientist may evaluate is its stack divided by the cost of checking one.

Ecologies as a fibration.

A configuration is a partition of \(\Theta\) across loci together with a structural assignment; an ecology is a configuration with a distribution over reachable configurations. The assignment of ecologies to theories is a fibration rather than a functor, with a canonical section given by taking cost as the weight map, and functoriality requires logic-preserving morphisms because the senses are structural §21.15 of \(\chSci\). The present note is largely an argument that this fibration wants a tensor.

23.3 The learner

23.3.1 Three layers, and the distribution is one of them

What we have been calling a learner has three components, and it is worth separating them before asking what composition does, because composition acts differently on each.

1. A population. A term: a parallel composition of scientists, of mortal computations holding no hypothesis, and of reservoirs. This is an object of the calculus.

2. A namespace. The family of names the population carries—metabolic channels, source channels, probe channels, mating channels—described by a name predicate rather than listed, which by Proposition 23.1 is available because the names are manufactured from material the population already holds.

3. A distribution. A weighting on the configurations reachable from the population, in the sense of §21.13 of \(\chSci\): the microcanonical ensemble at fixed \(\Theta\), sampled by the construction of Chapter 13.

We take the third to be part of the learner rather than a decoration on it. The alternative—a learner is the population, and the distribution is a decoration indexed over it—would put composition in the base of the fibration of §21.15 of \(\chSci\), where it is uninformative: two populations composed in the base are just two populations. Putting the distribution inside the learner puts composition in the total space, which is where the interaction lives.

23.4 Composition

23.4.1 One operator, three actions

On the first component the operator is the calculus’ own parallel composition: associative and commutative up to structural congruence, free, and adding nothing. On the second it is disjunction of predicates, which is again free, and is where describing rather than enumerating pays—one does not have to compute a union of two sets of names, one takes an \(\vee\) of two formulae. On the third the operator is not determined by its arguments at all. That is the entire subject of this section.

That \(\delta\) is a property of the pair is the first sign that composition is not going to be innocent, and §23.9 makes it a measurement.

23.4.2 Disjointness is factorization

23.4.3 The lax structure map is the interaction

Composition of learners is a candidate monoidal structure on the total space of the fibration \(\pi : \mathrm{Eco} \to \mathrm{GSLT}_{\mathcal{C}}\) of §21.15 of \(\chSci\), with unit the empty learner \((\mathbf{0}, \bot, \delta_{\varnothing})\). The obvious question is whether the canonical section \(s\)—cost as the weight map—is monoidal, and the answer is no, in a way we think is more useful than a yes.

23.5 Typed overlap: the grading vector

23.5.1 One grade is not enough

The preceding chapter’s Proposition 22.2 is the lesson in miniature. There the connective was \(\sep{\Nsp}{0}\) and the concept was fork; grading over line names made the formula true of pairs of threats sharing a gap, which are not forks, and grading over square names made it exact. The choice of \(\Nsp\) was not a parameter of the encoding but the whole content of it.

At the level of learners the same demand recurs, and the namespace of a learner is not homogeneous. A shared metabolic channel and a shared probe channel are both “an overlap of one name” and they are not remotely the same relation.

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23.5.2 The relations, derived

The point of typing the overlap is that the classical table of ecological relations then falls out as sign conditions on the components, rather than being imported as a taxonomy. Write \(\Delta_i\) for the change in learner \(i\)’s expected residual life under composition relative to isolation.

This is the same argument as \(\chGame\), Proposition 22.2, one level up, and we take the recurrence as evidence that choosing the grading namespace is where the modeling work in this framework actually happens.

23.6 Cooperation and competition are structurally unlike

23.6.1 Two currencies, one conservation law

Under Remark 21.10 of \(\chSci\) every metabolic transfer is zero-sum: predation is the purchase of somebody else’s remaining time. It follows immediately that cooperation cannot be a token transfer leaving both parties richer, and one might conclude that a conserved world has no room for mutualism at all. It does, and the room is in exactly one place.

So the only positive-sum overlap is epistemic, and it has two mechanisms, both already in the companion notes.

23.6.2 Cost relocation, and who is billed

Remark 22.9 of \(\chGame\) makes the point that putting threats into the term relocates a cost from the scientist’s budget to the world’s, and that the interesting quantity is not whether the work is done but who is billed for it. Composition is what makes that question have two answers inside one system. A shared legible surface is stigmergy, and it is the only mechanism by which two mortal learners in a conserved world can both come out ahead.

23.7 Non-interference is not stable

Disjointness of namespaces is the condition under which composition is innocent, so it matters whether it can be relied upon. It cannot, by default, and the reason is one the companion note already records for a different purpose.

The discipline exists, it is cheap, and it is the one the board of §22.4 of \(\chGame\) already uses.

23.8 Merger

Overlap in the mating namespace is not a kind of interaction. It is the failure of the two learners to be two.

Merger and composition agree on the first two components of Definition 23.1 and differ on the third, which is precisely why the third had to be part of the learner. As operators they behave differently in ways worth recording.

23.9 Phase is not compositional

Figure 2 of \(\chSci\) divides the \((\iota,\delta)\) plane into three regions: at low \(\delta\) food lies in the open and senses suffice, so a theory is overhead; at high \(\iota\) or high \(\delta\) discovery costs more than it returns and everything starves; in between, hypothesis formation pays. The boundaries are where the foraging inequality changes sign.

The interesting case is none of these three, because all three are couplings a modeller can see coming. The case worth measuring is the one in which the two learners never perceive each other at all.

23.10 Competitive dilution, measured

23.10.1 The model, and its one choice

Take \(\mathbf{k} = (0,1,0,0)\): two learners sharing exactly one source name and nothing else. Neither can perceive the other—no probe name is shared, so no assay of either reaches the other’s surface—neither can harvest the other, and neither can interbreed with it. By Table 23.1 the relation is exploitative competition, and the only channel of influence is \(\Theta\) itself.

The source is a single contract, hence rate-limited and non-exclusive (\(\chSci\), Definition 21.11). We must say how the two learners’ calls interleave, and this is the note’s one genuinely free modeling choice.

The alternative—equal shares regardless of cost—is defensible if one reads the contract as serving callers round-robin rather than first-come. We work Definition 23.6 and note in §23.14 what changes under the alternative: the dilution survives, the asymmetry does not.

23.11 The worked example: two games, one wall

23.11.1 A second environment

The game note laddered noughts-and-crosses and found the complete theory dominated. To compose we need a second learner, and the second environment should differ in the one respect that matters: whether its complete theory is affordable.

We take Nim, normal play [55]. The ladder is built by the same method and priced by the same schedule \(\kappa(d) = 2^d\), with the cost of a decision at rung \(r\) equal to \(|M| \cdot \sum_{\text{rules } 1..r} \kappa(d_{\text{rule}})\) and \(|M|\) the number of legal moves.

Rung 1 says: exactly one heap is nonempty, take it. Rung 2 says: exactly two heaps are nonempty, equalize them. Rung 3 is Bouton’s theorem: move to a position whose nim-sum is zero. All three are predicates on the heap sizes, read off the reflected structure of the position by a name predicate; none contains a modality, and rung 3 in particular requires no lookahead at all.

23.11.2 The ladder pays, and here it keeps paying

All figures are exact over all plays, symmetrized over which side moves first, against a uniform-random opponent, computed by memoized recursion (Appendix 23.17).

This is worth stating carefully because it is the hinge of the composition argument. Remark 21.9 of \(\chSci\)—yield, not truth—says a lineage under selection drifts toward yield-biased search wherever yield and discrimination disagree. Table 23.2 shows they do not always disagree. Whether they do is fixed by the environment: by how much of its regularity is structural rather than reduction-following (Remark 21.6 there), and by the branching factor, which by Proposition 22.3 of \(\chGame\) is the arity of the quantification over redex positions. So is truth affordable? is not a question about scientists. It is a question about worlds, and composition is the operator that puts two answers into one budget.

23.11.3 The composite

Let \(\Lrn_A\) be a population playing noughts-and-crosses and \(\Lrn_B\) a population playing Nim\((1,2,3)\), each against the practice wall of §22.3.2 of \(\chGame\), rooted at distinct quotations so that Proposition 23.9 applies. They share exactly the wall: \(\mathbf{k} = (0,1,0,0)\). The wall’s reserve is \(R_0 = 12{,}000\) at quantum \(q = 6\), and the price is \(p = 0.05\) tokens per metered unit, as in that note’s world.

At \(p = 0.05\) and \(q = 6\) a rung is self-funding exactly when its expected metering is below \(q/p = 120\). For the noughts ladder that admits rungs \(0,1,2\) and excludes \(3,4,5,6\); so \(\Lrn_A\)’s deepest self-funding rung is rung 2, at \(c_A = 77.8\) and a net of \(+2.11\) tokens per assay, and every deeper rung must be funded out of accumulated surplus. For \(\Lrn_B\), \(\chi_{\mathrm{Nim}}\) costs \(c_B = 38.5\), net \(+4.08\): the complete theory of \(\Lrn_B\)’s environment is self-funding, and the incomplete theory of \(\Lrn_A\)’s environment is not.

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23.11.4 Succession is a change of the grading vector

The wall is finite, so this composite has an ending, and the ending is where the typing of the overlap earns its keep.

In the worked composite, \(\Lrn_B\) ends the source phase holding \(5452\) tokens and \(\Lrn_A\) holds \(1397\); \(\Lrn_B\)’s stack is \(3.90\) times \(\Lrn_A\)’s. By the foraging inequality that stack is worth opening at any concealment depth these prices make affordable. So the learner that was confined during the source phase is the one positioned to harvest at succession, and the capital it harvests was accumulated by the learner that confined it.

23.12 The three units of learning

Collecting the three notes, the units of learning are distinguished by exactly one thing: the revision distance each can reach.

23.13 Composition is the exit from the dead ecology

Proposition 22.7 of \(\chGame\) says a population all of whose members hold theories that draw against one another performs no harvest, burns tokens to live, and starves; and that the better its science the sooner it dies. Remark 22.12 there lists three exits and takes none of them: enlarge the game so \(\chi\) is unaffordable, let the rules drift, or admit an external source.

The first exit is composition.

23.14 What this shows, and what it does not

Shown.

That a learner may be taken to be a weighted population and hence a namespace, and that under that reading composition is parallel composition on terms, disjunction on namespaces, and undetermined on distributions (§23.3–§23.4). That disjointness of namespaces is exactly factorization of the distribution, so the lax comparison map’s defect is the interaction and is a quantity rather than a caveat (Theorem 23.1, Proposition 23.3). That a typed grading vector recovers the ecological relations as sign conditions (§23.5). That conservation makes epistemic overlap the only positive-sum overlap (§23.6). That non-interference is unstable by default and purchasable by rooting (§23.7). That phase is not compositional (§23.9) and that competitive dilution confines a learner without contact, in a closed form and on exact numbers (§23.10–§23.11).

Not shown.

We have not proved a monoidal-category theorem: Proposition 23.3 gives the comparison map and its invertibility condition, and the coherence conditions, the unit laws up to the identifications merger imposes, and the relationship to the fibration’s section are left open. Definition 23.6 is a modeling choice; under the round-robin alternative each learner receives half the payouts and the retained fraction becomes \(1/2\) for both, so the dilution survives and the regressiveness weakens to whatever asymmetry the rungs themselves carry—in the worked composite \(\Lrn_A\) would retain \(0.500\) rather than \(0.331\), and still lose depth where \(\Lrn_B\) loses none. As in \(\chGame\) the measurements are exact computations over policies rather than observations of a running population: the ecology of Appendix 23.16 is written and not run to a fixed point, so every claim about invasion and succession is a claim about payoffs. And the Nim ladders, like the noughts ladder, are priced by a schedule that is a choice.

The honest summary.

The composition operator was, going in, an obvious formality—parallel composition is already there, and saying that learners compose by it looks like a restatement. What the exercise produced instead was three things we did not put in: that the probabilistic content of composition is exactly a graded separation, that a typed grading vector is forced rather than convenient, and that two learners which cannot perceive each other can nonetheless confine each other’s inquiry by a measurable amount.

23.15 Open problems

1. Coherence. Is \((\mathrm{Eco}, \mid, \mathbf{0})\) lax monoidal with \(\lambda\) of Proposition 23.3 as the structure map, and is a section of \(\chSci\), Remark 21.25, lax monoidal over it? Since that remark withdraws the claim that any one section is canonical, the question now has a second half: does the answer depend on which monotone cost-to-rate map was chosen, and if it does not, that independence is itself the canonicity the earlier claim wanted. The obstruction to strictness is located; the coherence conditions are not checked.

2. Is the grading vector complete? Definition 23.3 gives four sub-namespaces because four are what the companion notes distinguish. Is there a fifth? The candidate is the arena namespace: the names by which an admitted context class \(\mathcal{A}\) is realized (\(\chGame\), Observation 22.1). Two learners sharing an arena but no other name play the same game against different opponents, which is neither of the relations in Table 23.1.

3. Optimal \(\mathbf{k}\). Given a distribution over environments, is there an optimal grading vector for a learner to hold—an amount of overlap that maximizes expected residual life? §23.6 says the \(\Prb\) component should be positive and the \(\Met\) component should be zero or pooled, but the trade-off against the exposure of Lemma 21.1 of \(\chSci\) is not worked.

4. Run the composite. Appendix 23.16 specifies a two-learner world; §23.11 computes payoffs. Sampling trajectories by the Gillespie construction of Chapter 13 would turn Table 23.3 from a payoff calculation into an ensemble result, and would settle whether the succession of Proposition 23.14 actually occurs or whether \(\Lrn_A\) starves first.

5. The plagiarism equilibrium, now well posed. Problem 2 of \(\chSci\) asked whether honest science survives theory theft. With the grading vector it becomes: for which \(k_{\mathrm{prb}}\) is the two-way reading equilibrium stable against a one-way reader, and does the answer depend on the shape of the ladder (Observation 23.4) in each learner’s environment?

6. Does the series continue? Individual, lineage, composite: each reaches a revision the one below cannot. Is there a fourth unit, and if so what does it reach? The natural candidate is a distribution over composites—a distribution over grading vectors rather than over configurations—and we do not know whether that is a new object or the same one at a coarser resolution.

7. Cheapness versus completeness. Observation 23.4 shows that whether \(\chi\) is a good purchase varies by environment. Is there a criterion—presumably in terms of Remark 21.6 of \(\chSci\), how much of the environment’s regularity is structural, together with the branching factor—that predicts the shape of the ladder without computing it? For a composite this would predict who dilutes whom in advance of any measurement.

23.16 Appendix: A composite world, in rholang

The listing below is the composite of §23.11: two learners rooted at distinct quotations, sharing exactly one source name, with the typed namespaces of Definition 23.3 written as name predicates and the grading vector as a family of graded separations over them. Guards using a structural or modal connective are marked //! ideal and specify a check rather than perform one; everything else runs. Conservation is auditable by inspection: Wall, Harvest and Beget move tokens, Live debits them, and nothing mints.

// ============================================================================
// TWO LEARNERS, ONE WALL
// ============================================================================
//
// Companion listing to "Composing Learners".  The world is literally
//
//     Wall  |  L_A  |  L_B
//
// and every claim of the note is a claim about what that bar does.
//
// ROOTING (Prop. "Rooting makes disjointness a theorem").  Each learner
// manufactures every private name by quoting a term containing its OWN root.
// L_A's names are @[*rootA, ...] and L_B's are @[*rootB, ...].  By freshness-
// by-quotation no name of one is structurally equal to a name of the other,
// so k = 0 holds on every component of the grading vector EXCEPT the wall,
// whose name neither learner manufactured and both were handed.  That single
// imported name is the whole of the coupling.
//
// TYPED NAMESPACES (Def. "Typed namespaces").  Four predicates per learner:
//
//   Met(r) := @[ *r, "m", _ ]      metabolic: where stacks rest
//   Src(r) := @[ *r, "s", _ ]      source: emitters drawn upon
//   Prb(r) := @[ *r, "p", _ ]      probe: where assays are installed
//   Mat(r) := @[ *r, "g", _ ]      mating: where genomes are disclosed
//
// The grading vector of the composite is then the four-tuple
//
//   k_tau = | fn(L_A) /\ fn(L_B) /\ N_tau |
//
// and the note's claim is that this world has k = (0,1,0,0).

new stdout(`rho:io:stdout`), display, wall, rootA, rootB,
    Live, Wall, Assay, Ladder, Body, Harvest, Beget, Merge, Learner
in {

  for (@line <= display) { stdout!(line) }

  // ==========================================================================
  // METABOLISM.  Reclaim - debit - reissue.  Between the receipt and the send
  // the stack is a message in flight, which is what Harvest exploits.
  // ==========================================================================
  | contract Live(m, @burn, step) = {
      for (@sigma <- m) {
        match sigma > burn {
          true  => { m!(sigma - burn) | *step | Live!(*m, burn, *step) }
          false => { display!(("DIED", "could not fund a step")) }
        }
      }
    }

  // ==========================================================================
  // THE SHARED SOURCE.  One contract, hence rate-limited; non-exclusive, since
  // anyone holding the name may call it; non-adaptive; and finite.
  //
  // This contract IS the entire overlap.  Note what it does NOT do: it does
  // not know who is calling, does not adapt to callers, and cannot be used by
  // one caller to observe another.  Competitive dilution is delivered by a
  // process with no model of anybody.
  // ==========================================================================
  | contract Wall(w, @quantum, @reserve) = {
      for (ret <- w) {
        match reserve >= quantum {
          true  => { ret!(quantum) | Wall!(*w, quantum, reserve - quantum) }
          false => { ret!(0)       | Wall!(*w, quantum, 0) }   // exhausted
        }
      }
    }

  // ==========================================================================
  // THE ASSAY.  Unchanged from the companion notes: the hypothesis sits in
  // GUARD position, so playing IS testing and confirmation IS the rendezvous.
  // The probe name is manufactured from the learner's own root, which is why
  // k_prb = 0 -- neither learner can install a guard that the other's activity
  // could ever satisfy, because the names do not meet.
  // ==========================================================================
  | contract Assay(r, @g, @hyp, ret) = {
      new probe in {
        @[*r, "p", g]!(*probe)                     // probe in Prb(r)
        | wall!(*probe)                            // the ONE shared name
        //! ideal: structural guards
        | for (@pos <- probe where pos |= hyp)    { ret!(("confirm", pos)) }
        | for (@pos <- probe where ~(pos |= hyp)) { ret!(("refute",  pos)) }
      }
    }

  // ==========================================================================
  // THE LADDERS.  Two environments, two shapes.  L_A's ladder is the
  // noughts-and-crosses ladder of the game note: chi is dominated and every
  // rung above 2 runs a metabolic deficit at the wall's quantum.  L_B's is the
  // Nim ladder: chi_Nim is a NAME-LEVEL predicate of depth zero, so the
  // complete theory of L_B's environment is self-funding.
  //
  //! ideal throughout.
  // ==========================================================================
  | contract Ladder(@which, @r, ret) = {
      match (which, r) {
        // -- L_A: noughts-and-crosses, costs 0 / 40.9 / 77.8 / ... / 679.2
        ("ttt", 2) => { ret!( @{ exists s. Emp(s) and
                                 (Thr(s, me) or Thr(s, them)) } ) }
        ("ttt", 5) => { ret!( @{ exists s. Emp(s) and
                                 (Thr(s, me) or Thr(s, them))
                                 or <<K s>> Fork(me)
                                 or forall t. <<K s>>[K t] ~Fork(them)
                                 or Central(s) } ) }
        ("ttt", 6) => { ret!( @{ nu X. forall s.
                                 [K s]( ~Lost and exists t. <<K t>> X ) } ) }
        // -- L_B: Nim.  Note the absence of any modality anywhere.
        ("nim", 1) => { ret!( @{ exists s. One and Takes(s) } ) }
        ("nim", 2) => { ret!( @{ exists s. (One and Takes(s))
                                 or (Two and Equalises(s)) } ) }
        ("nim", 3) => { ret!( @{ exists s. (One and Takes(s))
                                 or (Two and Equalises(s))
                                 or NimSumZero(s) } ) }        // chi_Nim
      }
    }

  // ==========================================================================
  // A LEARNER.  Not a scientist: a POPULATION, rooted, weighted by the stacks
  // its members hold.  What makes it one learner rather than several is the
  // root, which is what its whole namespace is generated from.
  // ==========================================================================
  | contract Learner(r, @which, @rung, @burn, @members) = {
      match members {
        [] => { Nil }
        [h ...t] => {
            Body!(*r, which, rung, burn, h)
          | Learner!(*r, which, rung, burn, t)
        }
      }
    }

  | contract Body(r, @which, @rung, @burn, @i) = {
      new hyp, score in {
          @[*r, "m", i]!(400)                       // the stack: in Met(r)
        | score!(0, 0)
        | Ladder!(which, rung, *hyp)
        | Live!(@[*r, "m", i], burn, @{
              new ret in {
                  for (@h <- hyp) { hyp!(h) | Assay!(*r, i, h, *ret) }
                | for (@o <- ret) {
                    for (@c, @f <- score) {
                      match o {
                        ("confirm", q) => { score!(c + 1, f)
                                            | for (@s <- @[*r,"m",i]) {
                                                @[*r,"m",i]!(s + q) } }
                        _              => { score!(c, f + 1) }
                      }
                    }
                  }
              }
          })
      }
    }

  // ==========================================================================
  // SUCCESSION.  When the wall returns 0 forever, the only credit left is in
  // another learner's Met namespace -- which is a namespace this learner has
  // never described, because k_prb = 0 kept it invisible.  Acquiring it is a
  // step, and the step is metered: a hypothesis over Met(rOther) must be
  // formed and installed.  That is the change of grading vector, and it is
  // available only to a learner that still carries the apparatus.
  // ==========================================================================
  | contract Harvest(@rSelf, @i, @rOther, ret) = {
      //! ideal -- the guard is a name predicate over the OTHER root's
      // metabolic namespace.  Forming it is what costs; firing it is a comm.
      for (@mPrey <- @[*rSelf, "p", "hunt"]
             where mPrey |= @{ Met(rOther) }) {
        for (@sigma <- @{mPrey}) {
          for (@own <- @[*rSelf, "m", i]) {
              @[*rSelf, "m", i]!(own + sigma)
            | ret!(("harvest", sigma))
            | display!(("HARVEST across roots", sigma))
          }
        }
      }
    }

  | contract Beget(m, @alleles, @endow, ret) = {
      for (@sigma <- m where sigma > endow + 40) {
          m!(sigma - endow)
        | match alleles {
            [w, rg] => {
              new childM, code in {
                  code!( @{ Body!(*childM, w, rg, 2 + rg, endow) } )  // inert
                | for (g <- code) { *g | childM!(endow) }             // metered
                | ret!(("born", w, rg))
              }
            }
          }
      }
    }

  // ==========================================================================
  // MERGER -- a DIFFERENT OPERATOR, and this is the point of the section.
  //
  // Composition is the bar.  It is free, it is associative and commutative,
  // and it leaves both learners identifiable afterwards.  Merger identifies
  // the two MATING namespaces, which is a step, is metered, and is
  // irreversible: after it there is no composition recovering the parts,
  // because alleles have crossed roots.
  //
  // Note that Merge is a CONTRACT and `|` is not.  That asymmetry is the
  // formal content of "merger is a separate operator".
  // ==========================================================================
  | contract Merge(@rA, @rB, ret) = {
      new rC in {
          // the merged learner's mating namespace admits homologous loci
          // across BOTH roots; this identification is what cannot be undone
          @[*rC, "g", "admits"]!([rA, rB])
        | ret!(*rC)
      }
    }

  // ==========================================================================
  // THE WORLD.   Theta = 12000 (wall) + 3 x 400 (L_A) + 3 x 400 (L_B) = 14400.
  //
  // The composite is the bar between the two Learner calls.  Delete either and
  // the other's numbers change -- by exactly the factor of Prop. "competitive
  // dilution", and by nothing else, since no other name is shared.
  // ==========================================================================
  | Wall!(*wall, 6, 12000)

  | Learner!(*rootA, "ttt", 2, 4, [0, 1, 2])   // noughts: rung 2, c = 77.8
  | Learner!(*rootB, "nim", 3, 5, [0, 1, 2])   // nim: chi_Nim, c = 38.5

  // --------------------------------------------------------------------------
  // WHAT TO WATCH
  //
  // * The overlap is ONE NAME.  `wall` is the only identifier appearing in
  //   both Learner subtrees.  Everything else is manufactured under a root.
  //   k = (0,1,0,0), and it stays there by construction, not by luck.
  //
  // * Nobody models anybody.  No guard anywhere mentions the other root.
  //   L_B has no hypothesis about L_A and would gain nothing from one.  The
  //   dilution happens anyway.
  //
  // * L_B finishes and L_A does not.  L_B holds the complete theory of its
  //   environment at burn 5; L_A holds an incomplete one at burn 4 and cannot
  //   afford the rungs it still wants.  The one still climbing is the one the
  //   competition costs.
  //
  // * And it ends.  When the wall returns 0, Assay's confirm guard stops
  //   crediting, Live keeps debiting, and the only remaining credit is inside
  //   the other root's Met namespace.  Reaching it is Harvest, and Harvest's
  //   guard is a hypothesis somebody has to be able to afford to form.
  // --------------------------------------------------------------------------
}

23.17 Appendix: Reproducing the numbers

All figures in §23.11 are exact computations, not samples. The noughts-and-crosses ladder figures—win rates \(0.437\), \(0.668\), \(0.798\), \(0.831\), \(0.836\), \(0.918\), \(0.873\) and expected meterings \(0.0\), \(40.9\), \(77.8\), \(134.9\), \(245.7\), \(255.9\), \(679.2\)—are quoted from Table 1 of \(\chGame\) and were not recomputed here.

The Nim ladders are computed by compose.py, which enumerates positions as sorted tuples of heap sizes, evaluates each rung by memoized recursion over all plays with uniform tie-breaking within a rung and a uniform-random opponent, and charges \(|M| \cdot \sum_{1..r} \kappa\) at each decision of the policy player with \(\kappa(d) = 2^d\) and depths \(1, 1, 0\) for the three rules. Probabilities are carried as exact rationals and converted to decimal only for display. Figures are symmetrized over which side moves first; the unsymmetrized figures are reported separately where the color asymmetry is the point.

The composite of §23.11 uses \(R_0 = 12{,}000\), \(q = 6\), \(p = 0.05\), shares by Definition 23.6, and the identity of Proposition 23.13; the computation is a few lines and is included in the same script for auditability rather than for difficulty.