Where Does Agency Live?

Agency and determinism is an old argument, and one reason it does not move is that both sides help themselves to the boundary of the agent as though it were given. Fix the boundary and the question becomes tractable in an uninteresting way; leave it floating and the question is unanswerable. The previous chapter supplies something better than a stipulation. If choice is a resource and it is consumed at apertures, then there is a non-arbitrary criterion for where to draw the line: draw it around the regions that carry no observable choice.

That criterion has consequences, and this chapter is mostly a matter of following them. The first is that the regions in question turn out to be small. If selection, predation and competition are non-confluent — and they are, almost by definition, since which variant fixes and which lineage splits is exactly an outcome where the choice matters — then the determinate islands are local and the sea of races is dense. Apertures are everywhere, and “where is the agent?” has many local answers rather than one global one.

The second consequence is the one i did not expect. Decorating each cut with the strength of choice it demands turns the factorization into a filtration: not one partition into islands and seas but a family of them, one per level of choice strength, coarsening as the level rises. An observer endowed with strength \(C\) sees the world island at \(C\). Which means the stratification is not a tower imposed on the world from outside; it is the level-sets of a single decorated term, and different observers are different thresholds on the same object. This is also, and not by coincidence, what consciousness turns out to be in the chapters that follow — not a threshold an agent crosses but a point it occupies, with the ordinal tower of §58 the chain-shaped shadow of the lattice that point really lives in.

The chapter ends on a question rather than a result, and on a worry. The question is whether the graded factorization survives reduction — whether an island can crack while it runs, which would be the formal content of one agent perturbing another. The worry is that if choice in nondeterministic interaction is how hypercomputation would show up physically, it would be very hard to see. The closing section argues that hard to see is not the same as invisible, and takes Bell as the precedent.

60.1 Choice and the drawing of boundaries

We can now turn the instrument on agency. If nondeterminism is localized at cuts and is a resource, then the natural place to look for an agent’s boundary is where its choices are live — which is to say, at the apertures of Definition 59.1, governed by the principle that not every cut is one.

This is already a fresh answer to the old question. The apertures are exactly the races; the determinate traffic between them is not where agency is decided, however much computation flows through it. And it locates interference precisely: a high-capacity agent reaches into a low-capacity one not everywhere they touch but only at the cuts where the low-capacity one has a live, observable choice that the high-capacity one can see how to resolve. The asymmetry of capacity — one agent several halting-oracles above another — becomes operative only through an aperture.

Two corrections to a tempting oversimplification, both of which we will need. First, the aperture is a property of the cut, not of either agent; nondeterminism is not preserved by \(\Par\), so a determinate whole can have wildly underdetermined internal cuts, and conversely. “An agent’s nondeterminism budget is its interface” is therefore not quite right: the budget belongs to the cut, and which cut one calls the boundary is a modeling decision. Second, because of this, a seam where choice is genuinely open may sit far from the boundary an observer would draw, and its effects may be felt at cuts the observer counts as internal. We turn to a setting where this plays out concretely.

60.2 The biological axis: Smith redraws the boundary

Smith and Morowitz read terrestrial life as a planetary process and ask after its interface to the rest of the universe [52]. On their account the agents that matter at that interface are the ones that consume energy from outside the biosphere — sunlight, the chemical disequilibria of hydrothermal vents — while a vast interior of agents consume one another, trading energy already captured. The energy-eaters are the actual boundary between (terrestrial) life and the cosmos, and Smith presses the point we want: that we may have to be individual-organism-blind and treat the entire biosphere as the agent. The naive ontology places the agency boundary at the organism — this bee, that bird — and Smith argues that on a thermodynamic criterion the boundary belongs somewhere else entirely, around the whole. For him the biosphere is the macro-component, unified by its single interface to the extra-biospheric energy source.

This boundary-redrawing is exactly the move we need, and it is the only thing we take from the analogy. We do not take Smith’s interior to be determinate. It plainly is not. Schematically, \[\textsf{Biosphere} \;\rcong\; \textsf{EnergyEaters} \Par \textsf{AgentEaters}, \quad \textsf{Universe} \;\rcong\; \textsf{Biosphere} \Par \textsf{RestOfUniverse},\] the seam to the cosmos being \(\textsf{Biosphere} \Par \textsf{RestOfUniverse}\) and EnergyEaters the sub-term whose business is that cut — where outside energy, untyped to any particular molecule (the sun does not address its photons to a chosen chlorophyll), enters as an underdetermined produce, a genuine aperture. But AgentEaters is full of meaningful races, exactly the ones §59.1 put on the table: predation contended (two predators, one prey), competition for scarce resource, and, most decisively, natural selection and speciation, which are non-confluent almost by definition — which variant fixes, which lineage splits, is precisely an outcome where the choice matters and the histories do not reconverge. Smith’s biosphere-agent is therefore not a macro-component in our sense. Its interior runs the deepest races biology has.

Those races are not new to this book. The chapters on ecology took the same distinction — organisms that eat only external energy against organisms that eat one another — and derived it as an access profile rather than a kind of substance (Definition 21.11 and Proposition 21.12 of \(\chSci\)), with contended predation as a race on a metabolic channel and selection as the mechanism by which a lineage reaches what no individual can (§21.12 of \(\chSci\)). That two developments starting from different premises — one from conserved tokens, one from confluence — arrive at the same seam is the kind of agreement worth noticing, and it is not a coincidence: a linear send that can be received by either of two harvesters is a contended channel, and the foraging race is the aperture.

So the relationship between Smith’s factorization and ours is not that we fill in his interior; it is that we propose a different criterion for the same kind of move.

This still gives the seam-versus-boundary phenomenon of §60.1 its biological face, but more carefully than a single clean interface would. The energy cut is an aperture; its bias propagates into a teeming interior that has plenty of its own apertures; and the genuine computational question — whether such a bias is, at any given depth, transmitted through a determinate region or absorbed into a fresh race — is answered locally, region by region, not once for the whole. To answer it at all we need the determinate regions named precisely, and they are smaller than Smith’s whole.

60.3 Decorating the cuts: the factorization, graded by choice

The companion development [77] assigns to the nondeterminism at a cut a choice type: the strength of choice principle — a degree in the Weihrauch lattice [79], equivalently an altitude in the stalk of §58 — required to resolve that cut’s race, fairly and correctly, taken modulo resource bisimilarity so that branching which reconverges costs nothing. Write \(\chi(c)\) for the choice type of a cut \(c\), and \(\bot\) for the bottom of the lattice: no choice, computable resolution. Then the two readings of §59.5 collapse into a single labeling, \[\chi(c) = \bot \quad\Longleftrightarrow\quad c \text{ is not an aperture,}\] a cut being interior to a determinate region exactly when its choice type is trivial — whether because the match is unique or because all matches are \(\rbisim\)-equal, proof-irrelevant. The macro-component factorization of §59.5 is read straight off the decoration: the islands are the maximal subterms whose internal cuts all carry \(\bot\), the seas the cuts that carry more.

Once the cuts are labeled, the binary distinction — \(\bot\) against more-than-\(\bot\) — is revealed as the bottom case of a graded family. Fix any level \(C\) in the lattice.

A \(C\)-island is determinate to an observer endowed with choice strength \(C\): such an observer resolves everything inside and sees no open choice, and must climb only at the seas, where the type exceeds \(C\). This is the exact form of a relativity the inquiry has circled from the start — “determinate relative to a higher resource” — now indexed by a definite level, the observer’s own altitude. Determinacy is not absolute; it is relative to the choice power brought to bear, and the threshold \(C\) is that power named.

This is where the flat ontology and the tower meet, and it is worth saying exactly how. In §58 the stratification was external: a tower indexed over a category of models, a tower projected onto a term from outside. Here it is internal. The filtration \(\{C\text{-islands}\}_{C}\) is the term’s own stratification, read off the decoration \(\chi\) of its cuts, not imposed on it. The tower is the decoration. And the flatness is precisely that there is one term carrying one labeling, of which the strata are the level-sets: different observers are different thresholds, each seeing a different islanding of the same decorated term. “The tower is a cleavage of a flat jumble” (§60.5) sharpens into the statement that the cleavage is \(\chi\), and the jumble is the single \(\chi\)-decorated term beneath all of its thresholds.

There is a second reading of the filtration worth recording, because it connects this part to the one before it. Proposition 21.4 of \(\chSci\) says an individual learner revising in small steps can never leave the ball it started in, and Corollary 21.1 says the poorer it is the smaller that ball. Read through the decoration, confinement is a statement about threshold: a learner whose budget affords only choice strength \(C\) sees a \(C\)-islanding of its world and cannot even formulate a hypothesis that would require distinguishing what lies inside one of its islands. Poverty is confining because poverty is a low coordinate. And the escape found there — that the unit of learning must move from the individual to the lineage to the composite — reads here as raising \(C\) by acquiring a stronger scheduler, which is why composition and merger were the operations that worked.

Two honesties temper the construction. First, the decoration is well defined on a static term only insofar as subject reduction holds ([77]; §60.6): if scheduling can climb the stalk, \(\chi(c)\) rises as the term runs, an interior cut can cross above \(C\) and split its island, and the filtration is not invariant under reduction. Perturbation, in this language, is an island cracking as it runs — the \(C\)-factorization refining dynamically because a boundary decision manufactured strength inside. The crux of §60.6 is therefore exactly the question whether the graded factorization is a reduction invariant. Second, \(\chi\) is a semantic invariant, not in general computable — Weihrauch degree is undecidable at large. But it is approximable from above by types: linear internal typing forces \(\chi = \bot\), session typing bounds it, finite contention keeps it constructive. The term assignment of the companion note is the effective approximation of the decoration, which is why a factorization defined semantically can nonetheless be certified syntactically.

60.4 Stratification versus the jumble

The natural ontology suggested by “choice imports computational power” is a tower of oracles: agents stratified by degree, the stronger able to reach down into the weaker. We want to register a different intuition and then say exactly what saves it. A universe made of computational phenomena may be not a neat tower but a jumble — agents of all stripes and capabilities interacting, as we see in the physical world, with the stratification only a framework that aids understanding, not the structure of the world. If that is the case, agents with strong oracles could interfere with the “choices” of agents less computationally endowed, but they do so as participants in a flat field, not as occupants of a higher floor.

Both pictures at once: a (bi)fibration.

The two readings need not compete. Let a total category \(\mathcal{E}\) be the jumble — every agent of every degree, with the cut-mediated maps between them — and let a projection to the degrees (Turing, Weihrauch, or an ordinal index) be the stratification. Reindexing along \(\dgr' \le \dgr\) — the cartesian, fibration direction — takes a degree-\(\dgr\) computation to its degree-\(\dgr'\) shadow, the view a weaker observer keeps after forgetting the oracle. That is the “stratification as aid to understanding” move made precise: the tower is a cleavage, a chosen gauge for describing a frame-independent jumble. Interference is the other direction, and for it one wants the reindexing functors to carry Lawvere adjoints \(\exists_f \dashv f^{*} \dashv \forall_f\). We record, as a conjecture rather than a theorem, the fit that motivates the formalization:

The load-bearing check is whether the fibers carry the limits and colimits to support both adjoints; that is where a formalization should begin.

What keeps the jumble from collapsing.

A flat ontology of unequal agents is only coherent if a single strong agent cannot simply dominate. Two design decisions of the source vocabulary are exactly the consistency conditions. No implicit interaction [76]: the store never reacts with itself; a reaction is triggered only by a cut that actually co-locates opposite polarities. Drop it, and a single halting-oracle agent on a universal channel homogenizes the field to the top degree — the jumble dies. Locality of the cut: interference can be written only where a channel is shared, and, in the path reading, only into the subspace below a held prefix. Position in the trie and degree together set the reach of interference. Add a cost discipline (the phlogiston of the source program) and reach becomes finite per unit resource. Locality plus cost is what keeps a multi-degree universe a jumble rather than a hierarchy that flattens upward. We state this as a slogan, because it is the most decision-relevant thing the vocabulary offers: no implicit interaction and the risk ledger are not bookkeeping; they are the consistency conditions for a flat ontology of unequal computational agents.

60.5 The flat ontology

The tower of §58 is a map, and we should not mistake it for the territory. It is a way to understand computational phenomena — to sort agents by capacity and read interference as reaching down a tower. But the actual universe of computational phenomena may be far less tidy, and biology is the reason to suspect so. In a single living cell, fast molecular dynamics — vibrational and electronic transitions down to the femtosecond — coexist with transcription and translation that unfold over seconds to minutes: many orders of magnitude apart, intimately coupled, in one small volume. The cell does not run its fast and slow processes in separate strata that communicate through a clean interface; it juxtaposes them, and the juxtaposition is the point. The physical universe at large is the same — scales and scopes wildly interleaved rather than cleanly layered.

We propose to take computation the same way. In the flat ontology, the universe of computational phenomena is the total space of the fibration without privileging the projection — a jumble of agents of wildly different computational capacity, several halting-oracles up beside fully deterministic \(\lambda\)-like agents, interacting directly at cuts. The tower is not the structure of the world; it is a cleavage, a chosen gauge for describing a frame-independent jumble, and the “stratification” is an artefact of which morphism we chose to project along — or, read internally, of which threshold in the filtration of §60.3 we choose to view the term at. The islands-in-a-sea-of-races picture of §59.5 is the same claim seen from the agency side: determinate regions are local and sparse, apertures pervade, and there is no clean global stratum at which all the choice has been pushed to one rim. Read this in both directions, as promised. Computation, taught by biology, stops expecting a neat hierarchy and starts expecting juxtaposition. Biology, taught by computation, gains a vocabulary for what the juxtaposition costs and where it can bite: a high-capacity agent can influence a low-capacity one, but — Principle 59.1 — only at an aperture, only where the lower agent has a live choice the higher one can resolve.

What keeps such a flat universe coherent, rather than collapsing to the top capacity? Two disciplines, both already in the MPC vocabulary. No implicit interaction: the store never reacts with itself; a reaction is triggered only by a cut that actually co-locates opposite polarities [76]. Drop it and a single oracle on a universal channel homogenizes the field upward; keep it and interference must be cut-triggered. Locality and cost: a strong agent can write only where it shares a channel — and, in the path reading, only into the subspace below a prefix it holds — and a cost discipline makes its reach finite per unit resource. Locality plus cost is what lets agents of disparate capacity share a world without the strongest simply absorbing the rest. These are not bookkeeping; they are the consistency conditions for a flat ontology of unequal agents — and they are exactly the conditions under which a small confluent island can persist amid the surrounding races without being dissolved by the strongest agent that happens to share one of its channels.

60.6 The hinge: is determinism stable under interference?

One question decides whether the picture is static or dynamic, and it is the question a formalization should resolve first. Everything above treats “determinate interior, nondeterministic boundary” as a property a system has. Whether it keeps the property when a boundary aperture fires and the decision propagates inward is not obvious — because confluence up to resource bisimilarity, unlike linearity, is not compositional, and because a deep interaction in the path semantics manufactures a fresh, deeper interaction whose confluence is a new obligation [76].

The two answers are both meaningful, and the choice between them is what biology and computation are jointly asking. If rigidity is stable, the factorization is robust: a boundary race fires, the decision descends as a forced cascade through a confluent island that has no opinion of its own, and “determinate interior / nondeterministic boundary” is a genuine structure theorem holding along the dynamics — a buffered developmental cascade absorbing an upstream choice into one canalized outcome. If rigidity is not stable, then boundary interference can crack open a previously determinate island, turning a confluent agent into a branching one by descending into it. That is not merely a defect; it is plausibly the formal content of perturbation — a high-capacity agent not only steering a low-capacity one within its existing freedom but manufacturing new freedom in it and then steering that. In the ontological register: the difference between a universe whose agents have fixed capacities and one in which capacity itself is something agents can do to one another. We do not know which holds, and the same confluence lemma settles it on both axes at once.

60.6.1 The crux, in proof-theoretic dress

The term assignment of \(\chChs\) gives this question a second face, and the two faces are the same lemma.

If subject reduction holds, the graded factorization is stable. Determinate interiors stay determinate under scheduling, and a scheduler’s imported power is fixed by its type once and for all. If it fails, the failure is perturbation: a high-capacity scheduler, reducing against a low-capacity system, manufactures new choice in it — a well-typed term reducing to an ill-typed one, which is the type-theoretic signature of one agent changing another’s capacity. That a single metatheorem settles both the agency question and the metatheory of the correspondence is the best evidence we have that the correspondence is the right one. \(\chChs\) develops the correspondence, and what it owes.

60.7 Open problems

We collect the load-bearing uncertainties, each a structural claim one can try to settle. (i) Make precise the family-of-pairings whose section is the schedule and prove the \(\mathrm{AC}\)-grading of §59.3, identifying which schedulers are confluent up to \(\rbisim\) (constructive, no imported oracle) and which import altitude in the stalk. (ii) Exhibit the fibration \(p\colon\Hyp\to\mathbf{GSLT}\) rigorously and test whether angelic and demonic resolution are the Lawvere adjoints \(\exists \dashv (-)^{*} \dashv \forall\) to reindexing — the decisive sub-question being whether the GSLT fibers carry the limits and colimits both adjoints require. (iii) Show “maximal rigid subsystem” (Definition 59.2) is well defined and that the three frontiers of Proposition 59.2 coincide, and establish that a confluent interior is resource-degree-transparent. (iv) Resolve Question 60.1 — the stability of rigidity under descent — proving the structure theorem if it holds, or characterizing perturbation as a property of the sub-system if it fails. (v) Settle the scale question of §59.5: is any biosphere-scale structure confluent up to \(\rbisim\), or is confluence strictly small-scale with all large structure race-dominated? If the latter, confirm that Smith’s thermodynamic macro-component and our determinacy macro-components cross rather than nest, and characterize each as the carrier of a different invariant — and, via the filtration of §60.3, determine whether they nonetheless nest at some higher level \(C^{*}\) (Remark 60.3). (vi) Test the candidate that some local confluent island sits at ecosystem scale against a concrete ecological model, and locate the apertures that bound it. (vii) Develop the choice-type decoration \(\chi\) and the filtration by choice strength of §60.3: prove the monotonicity of Proposition 60.1, establish that the \(C\)-factorization is a reduction invariant iff the term assignment of [77] satisfies subject reduction (so that “perturbation” is exactly an island cracking under reduction), and pin down the syntactic approximations (linear and session types as upper bounds on \(\chi\)) that make the semantic factorization effectively certifiable.

60.8 Coda: hard to see, and the precedent of Bell

If this picture is right — if, in the physical world, the surplus power of a high-capacity agent shows up as nothing more dramatic than the way a low-capacity agent’s nondeterministic interactions get resolved — then hypercomputation in nature would be extraordinarily hard to see, and the difficulty would be principled rather than incidental. A confluent region hides its choices by construction (§59.5): the resolution happens but reconverges, leaving no trace. And even at an aperture, a single resolution looks like nothing — one outcome among those the situation allowed, indistinguishable, taken alone, from a coin flip. Choice-as-resource emits no per-event signal. Whatever evidence there is cannot live in any single outcome; it must live in the correlations across outcomes. The signature of a resolution stronger than a local choice function is that separated apertures come out correlated in ways no local, independently-resolving assignment could produce. That is why the resource is hard to test — and it tells us exactly where to look.

It is, almost word for word, the lesson of Bell. Bell’s theorem shows that no theory in which each measurement outcome is fixed by local, pre-assigned values — no local choice function for the outcomes — can reproduce the correlations quantum mechanics predicts for separated measurements on an entangled pair [88]; and the inequalities that mark the boundary have been violated in the laboratory, culminating in loophole-free experiments [89]. Read in our vocabulary, a quantum measurement is an aperture: a cut whose outcome the local term does not fix. Bell is then a statement about correlated choice — that the joint resolution of two separated apertures carries a correlation no local scheduler, no locally-supplied choice function, can account for. The per-event view sees only randomness at each wing; the structure is in the correlation, precisely as the testability argument above predicts. And the correlations have a developed mathematics: the device-independent resource theory of nonlocality treats these correlations as a graded resource [90], and the sheaf-theoretic account of contextuality [91] — logic-and-computer-science apparatus, already in our bibliography — characterizes exactly the obstruction to a global, locally-consistent assignment. The bridge between the choice mathematics and the physics already has a pier on the far bank.

We are not claiming that Bell violations exhibit hypercomputation, or that nonlocal correlations compute anything uncomputable: they do not, and the resource that nonlocality supplies is not Turing degree. The precedent is methodological, and in spirit it is decisive. It establishes that correlated choice is a real, mathematically characterized, experimentally confirmed feature of the physical world — that a phenomenon which is invisible per-event, visible only in correlation, and naturally described as the nonlocal resolution of an otherwise-underdetermined interaction, is not a fantasy of testability but the best-confirmed strange fact in physics. A research posture that treats choice as a physical resource, and looks for its signature in correlations across apertures rather than in single events, has therefore already succeeded once, spectacularly. That is reason to think it could succeed again, for resources further up the hierarchy than nonlocality reaches.

This is the note we want to leave the reader with, and it points in both directions, as the whole essay has tried to. Bringing the mathematics of choice — the axiom of choice and its gradations, the Weihrauch lattice, the resource theories — into contact with a physics informed by computation and biology may be fertile for each. The mathematics gains a fresh set of intuitions and motivations: choice stops being only a question in the foundations of sets and becomes a calculus of physical and computational resource, with the strength of a choice principle reading as the altitude it buys in a hierarchy of capacity, and with empirical stakes. The physics gains a new set of tools: the category of models and its fibration of hypercomputation, choice-as-resource as the vertical coordinate, the aperture and the macro-component as a vocabulary for where in a system agency and capacity actually sit, and a flat ontology in which entities of wildly unequal computational power — a halting oracle and a \(\lambda\)-term — can be imagined to meet, as so much else in nature meets, at a single cut.

Acknowledgements.

This essay draws on the RSpace denotation of the rho calculus [75, 76] and the GSLT/OSLF program [41], and on Smith and Morowitz’s reading of the biosphere [52]; its conceptual claims are deliberately routed back to structural obligations where they can be settled.