Chapter 62
Consciousness, and the World It Comes With
The last four chapters built a lattice and argued that every agent occupies a point in it. This chapter says what occupying a point amounts to.
The claim is not that high positions are better at thinking. It is that agents at different positions think differently and inhabit different worlds, and that the second half of that sentence is the substantive one: the ontology an agent’s bisimulation generates at a point is strictly richer than the ontology generated at any point below it. Consciousness, in this framework, is not something an agent has more or less of. It is where the agent is, and where it is fixes what there is.
62.1 The proposal
Let \(\mathfrak{W}\) be the Weihrauch lattice and let \(\mathrm{pos}(A) \in \mathfrak{W}\) be the choice strength agent \(A\) can afford to resolve at its cuts, in the sense of Chapter 60. The consciousness of \(A\) is \(\mathrm{pos}(A)\). An agent is conscious at \(w\) when \(\mathrm{pos}(A) = w\): it can form and evaluate the queries that \(w\) resolves, and its world model reflects the bisimulation quotient \(\bisim_w\) available at \(w\).
An agent that resolves only what finite contention resolves — which earlier drafts of this book called sentient but not conscious — is an agent at the bottom of the lattice. It is not outside the structure. It is at the floor of it.
The tower of Chapter 58 is indexed by ordinals, and §58.5 has already said, in that chapter, that the index is wrong. Definition 62.1 takes the index from the lattice instead, and it says something the ordinal version cannot.
Every agent occupies a position, and the position is its consciousness. There is no threshold separating the conscious from the non-conscious, and there is no transfinite ladder of grades either; there is a lattice, and everything is somewhere in it. The question “is this system conscious?” does not dissolve into “at what level?” — it dissolves into where, and the answer is a point that need not be comparable with the point another agent occupies. Two agents can each resolve what the other cannot. An ordinal index cannot say that; a lattice coordinate says it without further apparatus.
The second thing the coordinate reading makes explicit is that an agent’s position is simultaneously the strength at which the world islands for it, in the sense of the graded factorization of Chapter 60. The distinctions an agent can resolve and the granularity of the world it finds itself in are not two facts that happen to correlate. They are one fact, reported once from the inside and once from the outside.
62.2 Consciousness is a resource of a place, not a property of an agent
Definition 62.1 attaches a lattice point to an agent, which makes it look like a property the agent owns. It is worth being explicit that it is not, because almost everything interesting in this chapter follows from its not being.
Let \(A\) and \(B\) be agents with \(\mathrm{pos}(A) = \mathrm{pos}(B) = w\). Then the class of scheduling problems \(A\) can resolve and the class \(B\) can resolve are the same class, and the bisimulation quotient each world model can distinguish to is the same quotient \(\bisim_w\).
Both statements are functions of \(w\) alone. The first is the definition of a Weihrauch degree; the second is the construction of \(\bisim_w\) from what the degree resolves, which mentions no agent.
Proposition 62.1 is nearly trivial to prove and its content is entirely in what it licenses one to say.
Consciousness in this framework is not a private glow that each agent carries around. It is an affordance of a position: what can be resolved there, hence what can be distinguished there, hence what exists there. Every agent standing at \(w\) draws on the same affordance, in the way that every organism in a habitat draws on the same light. Two agents at \(w\) inhabit the same world, in the strong sense that the inventory of things in it is the same inventory.
What is emphatically not shared is \(\suff\). Suffering, by Definition 26.4, is a reading an individual takes of its own reserves against its own setpoint, and two agents at the same lattice point can be in wholly different states of it. That is the cleanest form of the distinction this chapter is drawing: sentience is owned and consciousness is occupied. One is a fact about a learner’s account; the other is a fact about a neighborhood in a lattice which the learner happens to be standing in.
Part Part II treats a population as a distribution of learners over a namespace, and its notion of what several learners have in common is a shared reservoir. That is a resource in the ordinary sense: rival, exhaustible, and reduced by each learner’s draw.
The resource in Proposition 62.1 is not like that at all. \(A\)’s resolving a query does not consume \(B\)’s capacity to resolve it, and a crowd at \(w\) does not make \(w\) poorer. A framework whose only notion of sharing was the reservoir had no way to express a common possession of this kind, which is one reason this chapter is here and not there.
62.3 A different world, and strictly a richer one
We can now argue the claim the chapter opened with. Throughout this section, \(v < w\) is the lattice order: \(w\) resolves everything \(v\) resolves and something more. The old presentation of this material stated each result for an ordinal successor \(\alpha \to \alpha+1\); every one of them is really a statement about the order, and that is how they are stated here.
62.3.1 Finer bisimulation quotient
Let \(v < w\) in \(\mathfrak{W}\) and let \(\GSLT\) be Turing complete. Then \(\bisim_w \subsetneq \bisim_v\) as equivalence relations on terms: there exist \(P, Q\) with \(P \bisim_v Q\) and \(P \not\bisim_w Q\).
By diagonalization. Since \(v < w\) there is a scheduling problem resolved at \(w\) and not at \(v\). Take \(P, Q\) whose distinguishability turns on the answer: their observable behaviors coincide on every experiment an agent at \(v\) can complete, and differ on one which requires that resolution. Then \(P \bisim_v Q\), and an agent at \(w\) separates them in one interaction.
The proposition says that the atoms of observable behavior are not the same at \(v\) and at \(w\). Terms that are experimentally indistinguishable at \(v\) — that no agent there can tell apart, however long it experiments — come apart at \(w\). There are more things in the world at \(w\). Not more information about the same things: more things.
62.3.2 Strictly more expressive hypothesis language
For \(v < w\), \(\HML(\ctx_v) \subsetneq \HML(\ctx_w)\): there are formulae expressible at \(w\) and not at \(v\).
Resolution at \(w\) gives rise to minimal contexts not present at \(v\) — those placing a term in interaction with what \(w\) resolves and \(v\) does not — and these generate modal operators absent from \(\HML(\ctx_v)\).
An agent at \(v\) not only cannot test certain hypotheses; it cannot form them. The concepts needed to think the thought are absent from the hypothesis language. This is the sense in which the two agents think differently: not more slowly and more quickly, but in different vocabularies, one of which contains words the other has no translation for.
62.3.3 New conserved quantities
Weight and cost maps at \(w\) can be sensitive to distinctions invisible at \(v\). Symmetries of the \(w\)-cost map which are not symmetries of the \(v\)-cost map give rise to conserved quantities at \(w\) with no analogue at \(v\).
In physical terms the world at \(w\) has strictly more conservation laws. What appears at \(v\) as one undifferentiated interaction resolves at \(w\) into a structured process with internal symmetries and corresponding currents. The analogy with the hierarchy of effective field theories is direct: new conservation laws — baryon number, lepton number, color charge — appear as one descends to finer scales.
62.3.4 Richer internal causal structure
A transition \(P \rewrite_v Q\) that is a single step at \(v\) may resolve at \(w\) into a structured history \(P = R_0 \rewrite_w \cdots \rewrite_w R_k = Q\) with internal causal structure visible only from \(w\).
This is the sharpest form of the richness claim, and it is worth stating without the mathematics. What looked like an atom of causal history at \(v\) has parts at \(w\). Not merely more of them: the things that were already there turn out to have insides. The analogy with the discovery that atoms have internal structure, and nuclei theirs, and protons theirs, is direct and intended.
And it is the fact this act will need in Chapter 65. An agent at \(v\) looking at \(w\)’s world does not see a world with things missing from it. It sees a world of the right size whose contents happen to be featureless, and it has no way, from inside, to tell that reading from the true one.
62.4 Sentience and consciousness
The distinction can now be stated exactly.
{1.4}
| Sentience | Consciousness | |
|---|---|---|
| What it is | A reading a learner takes | A position a learner occupies |
| Of what | Its account, against a setpoint | The choice strength it affords |
| Type | Signed vector over flavors | A point in a lattice |
| Owned or shared | Owned; private to the learner | Shared by all co-located |
| Changes | With inflow, outflow, and setpoint | By moving in the lattice |
| Analogue | Affect | The world one is in |
| Where built | Chapter 26 | This chapter |
An agent at the floor of the lattice can have a rich and violently changing \(\suff\). Its inflow fails, its drift is read, its policy shifts, and all of this is causally efficacious. But the world it lives in is the coarse world of the floor: it lacks the concepts to form the hypotheses available higher up, and those distinctions are simply not in its experience.
Conversely, nothing about standing high in the lattice makes an agent feel anything. An agent with no setpoint at all has \(\suff\) undefined and may sit anywhere. Nothing about being at the floor makes an agent numb; nothing about being high makes it tender. The two quantities are independent, and keeping them apart is the whole reason for two words.
Definition 62.1 does not require a setpoint, so the answer is formally yes. Whether it is stable is another matter: Conjecture 26.1 says internalization is selected wherever inflow is ragged, and a position high in the lattice is of no use to a lineage that starves before it can occupy one. Consciousness without sentience is possible in principle and, in any environment with variance, unlikely to persist.
62.5 The two cycles
The interaction between the two runs in both directions, and it is the reason they were ever confused.
The favorable cycle.
Resolution at \(w\) gives a finer world model. A finer world model gives sharper anticipation of inflow, which by Remark 35.1 is the entire point of paying for regulation. Sharper anticipation holds \(\vect{A}\) nearer \(\setpt\), so \(|\suff|\) falls and more of the account is free. A freer account funds the assays that keep resolution at \(w\) affordable — because by Chapter 60 occupying a position is not free either; it is a standing charge. And so back to the beginning.
The cycle is not guaranteed. It is broken by environmental disruption, by exhaustion, and by adversarial interaction with other agents, of which Part Part II supplies a great deal. What it describes is the stable attractor of a well-funded learner in a benign environment, and the name for that in another vocabulary is flourishing.
The unfavorable cycle.
Inflow falls. Drift grows, \(|\suff|\) grows with it, and by Proposition 26.3 the learner reallocates its search toward the components in deficit — which is correct, and which is paid for by giving up assays about the world. The world model coarsens. Anticipation degrades, so drift grows further. At some point the standing charge of the learner’s position stops being affordable, and the learner does not lose consciousness so much as descend: it comes to occupy a lower point, where fewer things exist.
This is the least comfortable consequence in the chapter and it should not be softened. A learner that can no longer afford its position does not merely become worse at seeing the world it was in. By Proposition 62.2 it comes to be in a different world, one with fewer things in it, and by Proposition 62.3 it loses the vocabulary in which the missing things could be missed. It does not experience the loss as loss. There is nothing left to notice it with.
The structural parallel to what is called cognitive decline is uncomfortably exact, and we make no clinical claim whatever.
62.6 Graded position
The lattice is not discrete, and an agent’s ability to resolve a scheduling problem may be partial.
An agent \(A\) has \(\epsilon\)-approximate resolution at \(w\) if for each instance of the scheduling problem characteristic of \(w\) it returns a correct answer with probability at least \(1-\epsilon\), where the probability is the one supplied by the stochastic instance of the weight construction of Chapter 13.
Earlier presentations of this definition measured the probability as \(|\langle \mathtt{halt} \mid A \inter \lceil (M,x) \rceil \rangle|^2\) — a squared amplitude, on the complex instance. That instance was proposed and withdrawn in Chapter 33, and by Proposition 13.3 an amplitude attached to a rewrite carries no coherence in any case. The quantity available is a rate and not an amplitude unless the presentation supplies a Hamiltonian, and Definition 62.2 is stated on that basis.
What is lost by the correction is the continuous interpolation the amplitude version advertised. A rate gives a probability of being right, which grades an agent’s reliability at \(w\); it does not by itself place the agent at a point strictly between \(v\) and \(w\). Whether approximate resolution defines a genuine intermediate position in \(\mathfrak{W}\) is open, and is the honest residue of a claim that used to be made too easily.