Tubular Agents

This part built learners at four sizes and then ran the architecture downward to a synapse. It has one thing left to do, and it is a thing that can only be done at the end, because it requires all of the machinery at once.

Return to Chapter 21 and to the distinction it drew in §21.11. A source is a persistent emitter: rate-limited, non-exclusive, non-adaptive. Prey is a linear send: exhaustible, exclusive, adaptive. Same sort, same rewrite rule, different multiplicity. That distinction was put to work immediately — it is what Corollary 21.5 rests on, and it is why plants do not need brains — and then it was set aside while the unit of learning moved four times.

We now have what Chapter 21 did not: an account of what an individual is (Definition 24.6), of which of its channels face outward (Definition 24.5), of what ownership of a pathway buys (Corollary 24.4), and of how one level’s outside becomes the next level’s inside (Proposition 24.6). With those in hand the trophic distinction says more than it did. It says something about shape.

The claim of this chapter is that an animal which must open its food is, topologically, a tube; that this is a consequence of an access profile rather than an observation imported from anatomy; and that the condition under which it holds can be stated as an inequality between two quantities the framework already meters. That last part is what makes the chapter worth ending a part with rather than a remark worth making in passing. A criterion that says when a tube is forced also says when one is not, and a claim that predicts bags as well as tubes has a test set rather than an illustration.

Much of what follows is conjecture and is labeled as such. The chapter is the part’s closing gesture rather than its last proof. But the gesture points at something checkable, and the checking has begun.

26.1 The observation

The reason for eating is to gain access to the token supply held inside the computation being consumed.

There is an advantage to minimizing the time spent exclusively on extraction. If some part of the extraction can be carried on internally — so that the consumer can meanwhile forage, mate, form hypotheses, and run assays — then extraction proceeds concurrently with everything else the learner does, and by the accounting of §21.7 that concurrency is worth paying for.

Eating therefore has two phases. There is a gross phase, which gets the relevant material across the boundary; and an internal phase, in which the remaining separation runs while the learner is engaged elsewhere.

The internal phase leaves a residue: the part that was not token. It must leave. Material in, refuse out.

26.2 The gut is an internalized enzyme

Chapter 24 put a converting medium between two learners and ranked media by behavior in Table 24.2: relay, lossy, corrupting, reading, converting, with the enzyme of Definition 24.2 at the top rung. The mycorrhizal network was the external instance of that profile.

This is not an analogy between two systems. It is one access profile appearing at two positions relative to a boundary, which is exactly what Proposition 24.6 licenses. The observation that makes the topology non-metaphorical is the next one, and it is a definitional consequence rather than a new claim.

A tube is therefore the construction that surrounds a region of the outside with the inside. It is the tower’s generating step applied for a specific purpose — to acquire boundary surface without acquiring exposure, in the sense of the exposure lemma of §21.9. The biological commonplace that the gut lumen is topologically external [244] is, on this reading, the same fact as the framework’s fractality, and not a coincidence to be noted alongside it.

26.3 Organs are communities

Here is where the part’s four moves pay off, and where the picture is not the one the trophic section of Chapter 21 could have drawn.

Chapter 23 made a composite out of two learners sharing a namespace, and Chapter 24 gave a criterion for where one individual ends and the next begins. Put those together with Observation 26.3 and something follows that is worth stating slowly. The boundary namespace of an individual is owned but open. Owned means it can be let. Open means whatever occupies it is not inside.

A digestive organ, on this reading, is not a specialized tissue that happens to convert. It is a community of learners tenanted in a boundary namespace, and the gradient of conversions along a gut — one flavor handed to the next — is a succession in the sense of §21.11, running in space rather than in time.

Two consequences make this more than a re-description.

The first is that it removes an objection. Chapter 24 noted that a market inside an individual is Coase’s theory of the firm [73] arriving unbidden, and that the arrival is awkward: if conversion is cheaper inside than across a boundary, why is there a boundary? Under Observation 26.4 the awkwardness dissolves, because the market is not inside. The rented enzymes sit in \(\Chn_{\partial}\), where pricing is still across a cut. The firm has an interior in which no market operates and a lumen in which one does, and the two are different namespaces.

The second is that it makes the herbivore’s symbionts the general case rather than a curiosity. By Theorem 24.4, tokens held in a foreign flavor cannot pass without an enzyme linking the flavors. A learner that eats far from itself must therefore either manufacture the conversions or rent them, and renting is what a boundary namespace is for. The rumen, the hindgut, and the microbiota that occupy them are not an accessory to digestion [247]; they are digestion, performed by a population that the host does not own but whose namespace it does.

Whether the letting can be done without the enclosure is a question this section cannot yet ask. It is the question §26.6 turns out to need, and Observation 26.5 is the answer.

26.4 What follows, and what would have to be shown

Two of the following are argued here. The rest are conjectures, and are labeled as such because the part’s standing commitment is that a claim be stated precisely enough to be wrong.

This is the argument of Chapter 22 one level up. There, a line signaling a threat on a square’s move channel would have rendezvoused with the square’s receipt and played the move; perception must not be able to act, and the separation was enforced by the choice of namespace. Here, egestion must not be able to feed, and the separation is enforced the same way.

What Proposition 26.1 does not say is when the alternation is cheap enough to live with. That is the business of §26.5, and it is where the chapter acquires its test set.

An earlier statement of this argument said autotrophs where Proposition 26.2 says non-phagotrophs, and the correction is not cosmetic. §26.6 is where it is forced, and it is the most useful thing the test set did.

In the source case, Proposition 26.2 and Corollary 21.5 are the same sentence about the same access profile, differing only in what is being purchased. One says that a source-feeder need not buy continuing inquiry. The other says it need not buy a topology. Plants do not need brains and plants are not tubes for one reason, not two. And the prediction is not merely that a plant needs no tube but that it should buy boundary area directly instead: a leaf and a root system are that purchase.

Conjecture 26.3 is the step that converts a remark about scheduling into a claim about topology, and it is the one we would most like to see settled.

Proposition 21.14 already says the overlap region between strategies is populated by composites rather than atoms, so a graded answer here is expected rather than disappointing.

The comparative anatomy is corroborating and instructively imperfect. Carnivores eat close relatives and have short, simple guts; herbivores eat far ones and have long ones with large fermentation chambers. But the association resisted statistical demonstration for a long time, and when it was finally shown across \(519\) mammal species the effect turned out to sit in the large intestine rather than the small, to be inseparable from phylogeny, and to admit enough scatter that the authors decline to call it a fixed law [246]. A framework predicting a monotone relation between flavor distance and stage count must account for that scatter, and the honest reading is that different morphological solutions exist for the same conversion problem — which is what Proposition 21.14 would lead one to expect.

If Conjecture 26.6 goes through it is the strongest result available here, because it makes a gross anatomical fact a consequence of an access profile. Intelligence and the tube become siblings rather than neighbors.

26.5 Rate of exchange: why breathing is not eating

Respiration is also an exchange across a boundary. Something is admitted, a fraction of it is extracted, and the remains are expelled. Yet the mammalian lung is a bag and not a tube.

The obvious thought is that the discriminating variable is rate: digestion runs orders of magnitude slower than the respiratory cycle, and it is the slowness that forces the second opening. That thought is pointing at something real. Rate as such is nevertheless the wrong variable, and saying why sharpens the chapter rather than weakening it.

26.5.1 Respiration goes tube where one would not expect it

Ventilation is unidirectional in fish — water in at the mouth, out at the opercular openings — and unidirectional in birds, where air moves the same way through the parabronchi during both phases of the cycle. Unidirectional pulmonary flow is not confined to birds. It has been demonstrated in alligators [265] and in the savannah monitor [266], and Farmer’s review argues from that distribution that the trait is ancestral for diapsids and therefore not an adaptation for high rates of gas exchange, since the animals that have it include ectotherms that do not fly; the candidate functions offered instead are reducing the work of breathing, evaporative water loss, and heat loss [264].

So a respiratory system can be a tube while running fast, and a bag while running fast. High rate is neither necessary nor sufficient. What varies across those cases is the ratio of demand to what a tidal exchanger can supply — water holds a small fraction of air’s oxygen and is far more viscous — rather than the rate itself.

26.5.2 Two parameters, not one

\(\Lambda\) is what sets the sequential fraction \(\sigma\) of Conjecture 26.2. If the control loop is blocked for the whole residence then \(\sigma = \Lambda / (1 + \Lambda)\), and the Amdahl-shaped bound bites only as \(\Lambda\) approaches and passes unity. This is the sense in which the rate intuition is right: slowness matters, but only relative to the opportunity structure. A process that takes a day is cheap to serialize if the next load cannot arrive for a week.

26.5.3 Breathing fails on separability, not on rate

The respiratory residue fraction is close to one — almost all of the mass admitted is expelled again — and bag topology survives that because \(\mu \approx 1\). The mammalian lung in fact tolerates gross re-mixing. An inspired bolus on the order of a quarter of a liter mixes with something like a liter and a half of residual gas, six parts spent medium to one part fresh, every cycle [264]. It works because the residue is the same phase as the resource and separation happens by diffusion at the membrane rather than by transport through the aperture.

Digestion has \(\mu \approx 0\). A bolus of partly extracted material cannot diffuse past incoming food, and the cost of re-mixing is not a few percent of exchange efficiency but contamination of the batch.

Proposition 26.3 subsumes Proposition 26.1 rather than replacing it. The mutual exclusion on a shared name is what the agent pays; \(\Lambda\) is how much of the cycle that exclusion consumes; \(\mu\) is whether the exclusion can be evaded by letting the two flows share the aperture after all.

The criterion is worth having because it has three clauses and each can fail separately. A theory that only ever predicts tubes explains nothing. This one predicts a bag whenever the agent is an osmotroph, or whenever loads arrive rarely relative to how long they take to process, or whenever the residue is the same stuff as the intake. Those are three different kinds of animal, and the next section asks whether they are the right three.

26.6 The test set

M. Stay proposed a list of animals with bag-like rather than tube-like digestive systems — cnidarians, ctenophores, flatworms, xenacoelomorphs and ophiuroids — together with two cases from outside the animals: fungi, which are heterotrophs, and plants such as Monotropa uniflora, which parasitize fungi rather than photosynthesize. It is the right test set, and it separates into three kinds of case: one that is factually mistaken, four that the occupancy ratio predicts, and two that force a correction of scope.

26.6.1 Ctenophora: the counterexample that fails

Comb jellies are not bags. Time-lapse imaging showed that the ctenophore gut is unidirectional and functionally tripartite, with waste expelled through terminal anal pores that are specialized to control outflow, resolving a long-standing misreading of those pores that had supported the blind-gut picture [267, 268].

This is more than the removal of a counterexample. Ctenophores branch very deep in the metazoan tree, so a through-gut there is evidence that the construction is not an inheritance of the bilaterian body plan but something arrived at from an access profile — which is this chapter’s thesis, and the strongest single piece of support it has.

26.6.2 Cnidaria: the predicted duty cycle

Mean digestion time in Aurelia aurita has been measured at about one hour [269]. Against the encounter interval of a drifting tentacle feeder on dilute plankton that is a low \(\Lambda\), and Proposition 26.3 accordingly predicts a bag.

The prediction with teeth is what should happen when \(\Lambda\) rises, and it has been observed. In Pelagia noctiluca ephyrae feeding on a dense and pulsed prey field, the gut saturates in about fifteen minutes while digestion takes about eighteen hours, and the authors read the mismatch as implying a diel feeding periodicity [270]. Fast fill, slow clear, forced alternation: that is Proposition 26.1 observed in a single organism, with the ratio measured.

26.6.3 Platyhelminthes and Xenacoelomorpha: area instead of length

Acoels have no gut lumen at all. The mouth opens into a syncytial digestive parenchyma without epithelial lining, in which particles are handled by phagocytosis. At body scales where the diffusion distance is a few cell diameters there is no bulk transport to arrange and \(\mu\) has no work to do. The branched blind gut of a triclad is the same move at larger scale: it buys boundary area rather than a second opening, which is the purchase Proposition 26.2 assigns to the non-phagotroph.

Cestodes are the sharper case. Tapeworms have no digestive system whatever and take up small organic monomers across the tegument. A cestode is an animal, a bilaterian, a heterotroph, and descended from ancestors with a through-gut — and it is not a tube, because something else did the digestion. No formulation in terms of heterotrophy can accommodate that. Proposition 26.3 accommodates it by the phagotrophy clause, and this is the case that forced the clause.

26.6.4 Ophiuroidea: secondary loss under a diet shift

Brittle stars lack an anus, as do paxillosid asteroids, and the review literature catalogues repeated independent losses of the anal opening across Bilateria, of which these are two [271, 272]. The direction of change matters: this is loss from a through-gut ancestor rather than a lineage that never built one, and it is concentrated in microphagous habits — suspension and deposit feeding on small particles, where the indigestible fraction per load is small and \(\Lambda\) correspondingly low.

That yields a within-clade test, which is the kind worth wanting. Macrophagous ophiuroids — scavengers and predators taking large items — should show markedly harder duty-cycling than their microphagous congeners, and if they do not, Proposition 26.3 is in trouble on ground of its own choosing.

26.6.5 Fungi and mycoheterotrophs: the scope was stated too widely

These are not counterexamples. They are a correction, and a useful one, because they identify the clause the argument had left implicit.

Heterotrophy is a claim about carbon source. The tube argument is a claim about phagotrophy: about admitting bulk material that contains something which will not be absorbed. A fungus secretes its enzymes outward and takes up monomers; the residue never crosses the boundary at all. Mycoheterotrophic plants do the same at one further remove, drawing fixed carbon through a mycorrhizal interface [273, 274], and Monotropa uniflora is the resolutive case, taking everything it has through associations with Russulaceae.

Observation 26.5 is what §26.3 was reaching for and could not yet say. There, letting a boundary namespace meant enclosure plus tenancy. Here it is tenancy without enclosure. The fungus is the tube construction turned inside out, which is why the mycorrhizal network appeared in Chapter 24 as the external instance of exactly the profile Observation 26.2 draws internally — and a leaf and a root system are the same purchase made by a non-phagotroph that is not an osmotroph either.

The cost structure is then legible rather than anomalous. Internalizing the chain buys exclusivity over a partially extracted harvest, which is Conjecture 26.3; externalizing it surrenders exclusivity and in exchange never has to transport residue. An osmotroph is an agent that has taken the other side of that trade, and it should be vulnerable exactly where the trade is worst — to theft of extracellular product by neighbors.

26.6.6 A prior statement, and what is left to claim

Honesty about priority. The functional claim is already in the literature: the same review that catalogues the losses states in passing that a one-way gut processes food more efficiently and permits uptake while the animal is still digesting [271].

What is claimed here is therefore not the observation but the derivation — that the two openings follow from concurrency together with ownership, and that the threshold is a computable function of \(\Lambda\) and \(\mu\) rather than a qualitative preference. Whether the derivation goes through is Conjecture 26.3 and the formal work below.

26.7 What would have to be written down

The two-phase harvest as a term.

An exclusive acquisition rendezvous on a boundary channel, in parallel with a persistent conversion chain on owned channels, terminating in a send on the expulsion channel. The Forage of the worked ecosystem in Chapter 21 and the chain of Chapter 24 supply most of it.

The tube as a namespace shape.

\(\Chn_{\partial}(B)\) factors as \(\Chn_{\mathrm{in}}^{\partial} \sqcup \Chn_{\mathrm{out}}^{\partial}\) with a directed redex pathway from intake to expulsion and no return path. Then tube is a formula in the generated logic rather than a designation, which is the move already made for individuality itself in Definition 24.6.

The occupancy ratio as a term-level quantity.

\(\Lambda\) should be recoverable from the harvest term itself: \(\tau_{\mathrm{proc}}\) as the metered cost of the conversion chain, \(\tau_{\mathrm{enc}}\) as the expected waiting time on the intake channel under the ambient offer rate. Then Proposition 26.3 becomes an inequality between two quantities the framework already meters, and the threshold is derived rather than posited. This is the single most valuable item on the list.

Separability as a namespace condition.

\(\mu\) is not a physical parameter but a statement about whether residue and intake can share a name without rendezvous — that is, whether the sorts involved make the crossed redex impossible. Gas exchange has \(\mu = 1\) because the spent medium and the resource are the same sort and the extraction happens at the wall rather than at the aperture. Stated as a condition on the sort discipline, Proposition 26.3 becomes entirely internal.

Whether the absence of a return path is forced.

It is not. Rumination and caecotrophy are return paths, and they occur exactly where one pass leaves too much unextracted: lagomorphs and hystricomorph rodents run a colonic separation mechanism that diverts fine particles and microbes back to the caecum and then re-ingest the product [248]. This should be stated as a condition on residual yield rather than suppressed, since a proposition that forbids observed behavior is worse than one that predicts it.

The physical topology as a shadow.

A through-gut makes a body a genus-one surface. That is real, but it is not the content. The content is the channel topology; the surface is what an embodiment of that topology in three-space looks like. Saying so explicitly keeps the argument from appearing to run from anatomy to mathematics, which is the direction it does not run.

26.8 Open questions

26.9 What is claimed

Claimed: that the residue of a harvest is a consequence of prey being computations rather than an added postulate (Observation 26.1); that a lumen is boundary namespace and therefore not interior (Observation 26.3); that a digestive organ is a community tenanted in that namespace, which is why a market can operate inside a firm without contradicting the firm (Observation 26.4); that an osmotroph is the same construction everted, tenancy without enclosure (Observation 26.5); and that intake and expulsion on one name force an alternation (Proposition 26.1).

Claimed with a different kind of support: the tube criterion, Proposition 26.3. It is not derived. What stands behind it is the test set of §26.6 — seven cases proposed as counterexamples, of which one was factually wrong, four came out as the occupancy ratio predicts, and two corrected the scope of the claim in a way that made it cover cestodes, which no formulation in terms of heterotrophy could have done. That is the support a criterion can have before the derivation exists, and the honest description of it is that the criterion has survived the first attempt to kill it.

Not claimed: the six conjectures, and in particular not Conjecture 26.3, on which any derivation of the criterion depends.

What the chapter does claim, and would defend, is a change in what kind of thing an anatomy is. Part Part II has spent its length arguing that a learner is a term among terms, paying; that the unit of learning moves; and that where one individual ends is a question with an answer in the framework rather than a matter of convention. If that is right, then the shape of an animal is not a separate subject that biology happens to study alongside cognition. It is another entry on the same bill. A learner that must open its food pays for inquiry, and pays again for somewhere to put the opening — and the second payment, made in surface rather than in tokens, is the reason that almost everything which thinks is also, in the end, a tube.