“Traces of Consciousness” — explained

1. What a Markov kernel is, in plain language

Everything in the pre-print — not yet peer-reviewed — is built out of one object, and it is much simpler than its name suggests.

A Markov kernel is a transition rule with amnesia. Given where a system is right now, it tells you the probabilities of where it goes next — and how it arrived at the current state is irrelevant. That is the whole idea.

Weather is the standard example. Suppose: if today is rainy, tomorrow is rainy with probability 0.6 and sunny with probability 0.4; if today is sunny, tomorrow is rainy with probability 0.2 and sunny with probability 0.8. That little table — four numbers — is a Markov kernel. It is Markov because the forecast depends only on today, not on the whole preceding week. It is called a kernel rather than a matrix only because mathematicians need the idea to keep working when the states form a continuum rather than a short list, which requires measure theory to state carefully. No deeper mystery is hiding in the word.

Hoffman's conscious agent is three such tables chained in a loop:

P (perceive) — from the state of the world to an experience.
D (decide) — from an experience to an action.
A (act) — from an action back to a new state of the world.

Compose the three and you get what the paper calls the qualia kernel, Q: the probability that your next experience is e′, given that your current experience is e. An agent, formally, is a space of possible experiences, a space of possible actions, a space of world-states, those three kernels, and a counter that ticks off which experience you are on.

The crucial thing to notice is what is not in that definition: any physical substrate whatsoever. There is no brain, no body, no matter. Consciousness is not something the mathematics produces — it is what the mathematics is made of. The authors are candid about the cost of this. Asked what makes experience possible in the first place, the paper answers: “The theory does not say. That is a ‘miracle’ of the theory.”

2. The one genuinely new idea: observation as a trace

Here is where the paper earns its title.

Imagine a process moving among many states, but you can only see some of them. You watch, and you record only the visible states, deleting the stretches when the process wandered through the ones you cannot see. What you end up with is a smaller process, defined only on the states you can observe. That reduced process is called the trace.

An analogy: a colleague's calendar shows their meetings but not their commute, lunch, or thinking time. From your side, they blink from meeting to meeting. Your view is a genuine, coherent description of their day — the trace of it — but it is a compression of a richer process, and the transition probabilities in your version are not the ones in theirs. The paper works a small example where a probability of 0.2 in the full process becomes 0.42 in the trace. Nothing is fabricated; the number changes because the hidden excursions have been folded in.

The definition the paper builds on this is short: agent A observes agent B if and only if A's kernel is a trace of B's.

Two consequences follow, and they are the reason the paper exists.

First: the observer is inside what it observes. Not adjacent to it, not looking at it through glass — a compression of it. In the authors' words: “any observer is an integral part of what it observes. Observers are not aloof, objective, and negligible… This is a radical departure from standard notions of detached observers with paltry influence.”

Second: the distinction between observer and observed collapses. Anything you observe is itself observing whatever it is a trace of. There is no bottom to the stack and no privileged vantage. “So the distinction between observer and observed dissolves.”

This is the paper's answer to a complaint physics has carried for a century. Quantum mechanics needs an observer and never says what one is; calling it a “measuring apparatus” relabels the problem rather than solving it. Frank Wilczek called modelling the observer a “formidable project.” John Wheeler imagined reality built from countless “acts of observer-participancy” without ever formalising it. The trace theory is offered as that formalisation.

The picture in ordinary language

Strip away the measure theory and the view the theory is offering looks like this.

There is nothing but conscious agents. Not agents plus a world they inhabit — agents, and what agents look like to each other. Everything a physicist would call matter is, on this account, how some network of agents appears when compressed into the particular view another agent has of it.

Each agent runs a loop. Perceive, decide, act, and the acting changes what there is to perceive next, which changes what is worth deciding, and around again. Three kernels; the qualia kernel is what you get when you compose all three and ask only about the sequence of experiences. Nothing in the loop is a substance. It is a shape that keeps happening.

Different agents have genuinely different spaces of experience. This is in the formalism, not read into it — each agent carries its own measurable space of qualia. There is no requirement that two agents' experiences be commensurable, or translatable, or even that they overlap. Agents clustered into interacting groups can develop stable structure among themselves — their own regularities, their own geometry, their own physics — without that structure being available to agents outside the group.

A caution on vocabulary: the paper uses communities and communicating classes as technical terms from Markov-chain theory, deployed to model bound and confined particles. The broader reading — communities of agents developing whole alternative realities — is a natural extension of the framework and Hoffman gestures at it in interviews, but it is not what those words are doing in this document.

Why some perspectives are reachable and others are not. Here the useful intuition is a decoder. Open a photograph in an image viewer and you get a face; open the identical file in a text editor and you get pages of garbage. Same bits, same file, nothing missing — but one program's structure makes the data into something and the other's does not.

The formalism says something close to this about observers. Two agents have a common vantage from which both can be checked only if they are what the paper calls simultaneously verifiable, and most pairs are not. Boolean logic — ordinary and, or, not — works cleanly inside any one agent's world and breaks down between worlds (Theorem 4.12, above). Inaccessibility is not distance and not secrecy. It is the absence of a shared decoder.

Agents combine, and this is where the theory is least finished. A human being, on this reading, is not a single agent but a very large number of them merged into a composite with its own qualia kernel — and merging can run in both directions, upward into larger composites and downward into components. That is the combination problem, and it is the one philosophical problem the formalism genuinely addresses.

It is worth knowing that this is also precisely where the mathematics stops. The appendix works out when two observers have a common lower bound cleanly enough, but on the upward direction it prints an open question and, in one case, three literal question marks. Merging is the most philosophically interesting claim in the theory and the least settled part of the apparatus.

Three triads, and how well they actually line up

The perceive-decide-act structure is not new, and the resemblances are worth checking rather than asserting.

Federico Faggin names three irreducible properties of consciousness — knowing, choosing, feeling — and the trilogy treats them as structural. Two of the three map: knowing to perception, choosing to decision. The third does not. Feeling is not acting; feeling belongs on the side of qualia itself, and Hoffman's action kernel has no Faggin counterpart. A real family resemblance, not an identity, and it should not be claimed as one.

Kashmir Shaivism fits considerably better. The Trika tradition describes three śaktis, powers of consciousness: jñāna (knowing), icchā (will), kriyā (action). Set beside perception, decision, and action, the correspondence is close to exact — including the third term, which is where Faggin's parallel fails. Two systems separated by a thousand years and every possible difference of method arrive at the same three-part decomposition of what it takes to be a subject at all.

That is worth noticing and it is not worth over-reading. Convergence on a triad may indicate that the structure is real, or that there are only so many ways to carve an agent. The trilogy's position is that it is suggestive rather than probative — the same standard applied to every other convergence on this site.

One tension a careful reader of this site will notice. If reality consists only of conscious agents, then what we call inanimate matter is also a projection of them, and consciousness is everywhere in some degree. That sits badly with the position argued in Why biology? §5, where a substrate without autopoietic organisation is not low on the gradient of reception — it is not on the gradient at all. Hoffman's framework has no such gate; this one turns on it. The two views cannot both be right as stated, and the disagreement is the same one set out in §7 below, arriving from the other end.

3. What the paper actually proves

Three technical results, stated here without the machinery.

The trace of a trace is a trace (Theorem 2.4). This is the technical heart of the paper, and in plain language it says: if you are a compressed version of something that is itself a compressed version of something larger, then you are a compressed version of the larger thing directly. Observation composes. Watch a colleague's calendar; the colleague sees only part of their own organisation; the organisation sits inside a market. Your view is a legitimate trace of the market, not merely of the calendar. The proof runs through a small combinatorial lemma about geometric series of operators, and it is the piece everything else leans on.

The trace order is therefore a genuine partial order (Theorem 4.2). Every kernel is a trace of itself; if two are traces of each other they are the same kernel; and transitivity follows from Theorem 2.4. This is what makes “observes” a coherent relation rather than a metaphor — without it, the nesting of observers would not be well defined.

There is a closed formula for computing traces (Theorem 3.4). You do not have to simulate the hidden excursions; the compressed process can be calculated directly from the blocks of the original. Worked example from the appendix: a three-state process with transition probability 1/3 from the first state to itself becomes, when traced onto the first two states, a process with 2/3 in that position. Nothing is invented; the excursions through the invisible third state have been folded back in. The paper is careful to note this is not the same as simply renormalising the visible part of the matrix.

Stationarity survives tracing (Theorem 3.9). Let the full process settle into its long-run distribution; the trace's long-run distribution is exactly the restriction of it. This is the bridge from observation to belief, and its proof is four lines.

Observation and belief have the same structure (Corollary 4.4). The map from a kernel to its stationary measure is a logic homomorphism from trace order to Lebesgue order. Informally: the way observers nest inside each other and the way beliefs nest inside each other are the same shape.

Two honest notes on that last one. It is stated for irreducible kernels — a restricted class — so the abstract's unqualified version is broader than the corollary. And its proof is a single line, which is unusually terse for the result the abstract headlines. (The paper also has genuine cross-reference slippage: that one-line proof points at “theorem 3.8,” but 3.8 is an example; the relevant theorem is 3.9. Several equation references elsewhere are similarly off. This is sloppiness in a preprint, not a defect in the mathematics.)

A borrowed vocabulary worth noticing

When two observers do have a meet — a greatest common observer below both — the paper calls them simultaneously verifiable. When they have a join, it calls them compatible. That vocabulary is lifted straight from quantum logic, where compatible observables are the ones that can be measured together without disturbing each other.

The implication, if you take the framing seriously: most pairs of observers are not simultaneously verifiable at all. There is generally no common vantage from which two perspectives can both be checked. That is a strong claim about the structure of perspective itself, and the paper makes it almost in passing.

It makes a second one in the same register, and this one is stranger. Compatibility and simultaneous verifiability are, in the authors' word, intransitive. If you share a vantage with me, and I share one with a third party, it does not follow that you and the third party share anything at all. Common ground does not chain. Whatever community of perspective exists is local and does not propagate outward — which, if the framework is right about anything, is a formal statement of something most people already suspect about the difficulty of mutual understanding at scale.

When can two observers merge? The appendix has a beautiful partial answer

The combination problem — how separate subjects join into a larger one — is the philosophical prize here, and the appendix gets further with it than the main text lets on, while also failing more interestingly.

The setup is an algebra problem. Given two observers, find the larger observer that both are traces of. That reduces to nine unknown matrix blocks constrained by nine equations, and the authors say plainly: “We have not here explored general solutions. It is an open question whether solutions always exist and, if so, are unique.” So the combination problem has been reduced to a tractable-looking system and then not solved. That is progress and it is not a result.

But one special case is solved cleanly, and it is worth stating in plain language because the condition is interpretable. The unique solution exists when the two observers' worlds are connected only through what they share — when there is no back-channel from one to the other that bypasses their common ground. In the notation, two off-diagonal blocks vanish; in ordinary terms: two perspectives combine into one determinate larger perspective precisely when everything that links them passes through what they have in common. If there are hidden routes between them that skirt the shared region, the combination is no longer uniquely determined.

That is the most philosophically suggestive line in the appendix, and the paper spends one paragraph on it.

Two ways combination can fail outright. Even in the clean case, the solution only holds if a certain matrix has no negative entries — otherwise there is no larger observer at all. Combination is not merely hard to compute; it can be algebraically impossible, and the failure condition is concrete rather than mysterious. And on the belief side, when two perspectives are mutually singular — sharing no common ground whatever — there is not one minimal way to combine them but a one-parameter family of them, mutually incomparable, with no principled reason to prefer any. Merging two perspectives that overlap nowhere is not impossible; it is underdetermined, which for some purposes is worse.

Buried in the same remark is a further open question the authors flag and do not pursue: whether two observers with a common vantage might be traces of more than one larger observer. If they can, then there is no fact of the matter about which superordinate subject you belong to — you would be a component of several, with nothing selecting between them. For a theory whose central application is the combination of conscious subjects, that question is not a detail.

One elegant piece of structure that does come out cleanly: within a shared world, negation is well defined. Your complement is the same world traced onto everything you are not — and it is itself an observer. “Not-you” is not nothing; it is the rest, viewed as a subject. You and your complement have nothing in common and together reconstitute the whole.

Why the logic is “non-Boolean,” and why that is not the quantum kind

Ordinary logic is Boolean: every proposition has a negation, any two propositions have an “and” and an “or,” and there is a universal truth at the top. Quantum logic famously breaks one specific rule — distributivity.

The trace logic breaks something else, and it is worth being precise because the two are often conflated. It has no top element (no universal observer that contains all the others), no complements (there is no “everything except this observer”), and for most pairs of observers there is no “and” and no “or” at all — the operations simply are not defined.

What survives is locality, and the appendix makes this exact (Theorem 4.12). Fix any one kernel; the collection of everything that is a trace of it is a Boolean logic — meets exist, joins exist, and even negation comes back, defined as the trace of the parent on the states the original observer does not occupy. So Boolean structure is not absent. It is relative. It exists inside any single observer's world and fails between worlds.

That is a substantive claim about perspective, not a technicality. There is generally no vantage from which two arbitrary observers can both be checked. The paper's own term for the pairs that can is simultaneously verifiable, and for those that admit a join, compatible — vocabulary borrowed openly from quantum logic, where compatible observables are the ones measurable together without disturbing each other.

So: not quantum logic, and more radically incomplete than it. The authors say so directly — the structure is “more general” than the orthocomplemented modular lattices of quantum theory.

Two details in the appendix that repay attention

Bayes' rule falls out as a special case. On the belief side, the “and” of the Lebesgue logic turns out to contain ordinary Bayesian updating: P(A|B) = P(B|A)P(A)/P(B) appears as a particular instance of the Lebesgue conjunction. The rule everyone uses for revising belief in light of evidence is not bolted on — it is a corner of the structure. Whatever one concludes about the metaphysics, that is an elegant result.

And one genuinely perceptual prediction. The paper notes, almost in passing, that “the existence of probability measures that are not simultaneously verifiable corresponds in perception to multistable percepts.” Multistable percepts are the Necker cube, the duck-rabbit, the spinning dancer — figures where perception flips between readings and refuses to blend them. On this account that flipping is not a quirk of visual processing. It is what it looks like from the inside when two beliefs have no meet: you cannot hold both, so you alternate. This is the one place in the entire apparatus where the formalism touches something a person can check against their own experience in the next thirty seconds, and it is worth more than several pages of the mass-and-spin material.

4. What the paper proposes rather than proves

This is most of the paper, and the authors flag it honestly — nearly every claim in this section opens with “we propose.”

The programme is to derive physics from properties of Markov chains. The proposed dictionary:

Mass ← the entropy rate of a process (how much genuine novelty it generates per step).
Spin ← the determinant of its matrix.
Speed ← the inverse of how long it takes the process to get between states and back.
Energy and momentum ← the inverse of a periodicity parameter.
Position ← an index over the process's long-run outcomes; time ← the step counter.
Free, bound, and confined particles ← how the entropy rate splits into “kinetic” and “potential” parts.

They also sketch a derivation of the Heisenberg uncertainty principle from a sampling conflict: pinning down momentum requires watching the process for a long stretch, pinning down position requires watching for a single step, and you cannot do both.

None of this is derived. The strongest technical claim in this part of the paper is that a known theorem about Markov chains produces expressions identical in form to free-particle wavefunctions. That is a form-match, not a derivation of quantum mechanics, and the paper does not claim more than it has.

One number is worth quoting because the authors state it without flinching. To reach a mass ratio of about 1834 — roughly the proton-to-electron ratio — the scheme requires a matrix of dimension 10⁵⁵² × 10⁵⁵². For comparison, the observable universe holds on the order of 10⁸⁰ particles.

The spin proposal, completed in §7, is the most specific thing in the dictionary and worth stating because it is unusually committal. Take the unit hypercube of a communicating class and push it through the kernel; it deforms into a parallelepiped whose signed volume is the determinant. The authors call the result the C-spin, and propose that the spin number and spin axis of a particle are its projection into spacetime. Three cases, and they map onto the three spin values physics actually uses: a massless class (zero entropy rate) gives a hypercube with no preferred direction, only a signed volume of ±1 — spin 1; a massive class with non-zero volume gives a parallelepiped with a major axis — spin ½, the axis being the projection of the major axis, up or down according to the sign; a massive class with zero volume gives spin 0. Whether or not it survives, it is a real commitment rather than a gesture: it says which geometric feature becomes which observable.

Two framing claims from §8 that change the picture. First, physical projection is described as “an act of limited proto-observation: there are limitations in both the extent of states observed and in the number of moments constituting the observation.” Matter, on this account, is not what consciousness is but what a restricted view of it yields — the limitation is constitutive of materiality rather than incidental to it.

Second, and more striking: “Not all (we expect vanishingly few) conscious dynamics will thus project to the physical world.” On their own reading, the physical universe is a vanishingly thin slice of what exists. Everything physics studies is the small remainder that happens to admit a projection into spacetime. That is a large claim, made in one parenthesis, and it is arguably the most consequential sentence in the paper.

5. Does it solve the hard problem?

No — and read carefully, it does not claim to. It relocates the problem.

Physicalism starts with matter and owes an account of how experience arises from it. Nobody has produced one. Conscious agent theory starts with experience and owes an account of how matter arises from it — and the paper's wager is that this second debt is payable while the first is not. Their sharpest formulation: “How many experiences have physicalist theories explained? Zero. No physicalist theory explains any specific conscious experience.” And: “The problem for conscious agents appears to be technical and manageable… But the problem for physicalist theories appears to be principled. It simply cannot be solved.”

That is a strong claim and the support offered for it is weaker than the claim: a long quotation of Leibniz's mill argument from 1714. An intuition pump, not an impossibility proof. It may well be right. It is not demonstrated.

Meanwhile the theory's own foundation is explicitly unexplained. Qualia are a primitive; what makes them possible is, in the authors' own word, a miracle of the theory. They defend this with a general point that is fair as far as it goes — every theory rests on assumptions it cannot itself explain, so “science can offer no theory of everything, in the sense of a theory that explains its own assumptions.”

And in §8 they say all of this themselves, more cleanly than any summary could. They even raise the sharpest objection against their own position first — if the formalism succeeds at recovering physics, why call it a theory of consciousness at all rather than an exotic physical theory that happens to live outside spacetime? Their answer:

Strictly speaking, any theory taking consciousness as fundamental is not a theory of consciousness, but a theory from consciousness. CAT does not purport to explain consciousness in its essence. Rather than assuming any putative notions of what's fundamental in the physical world (a world that is apprehended, originally, only in consciousness), it takes essential aspects of consciousness as axiomatic, and on this foundation attempts, as scientific theory, to explain the nature of our embodied experience.

A theory from consciousness rather than of it. That is the honest description of what the paper does, it comes from the authors rather than from a critic, and it settles the question this section opened with. The hard problem is not solved. It is declined, deliberately and with the reasons given.

What the formalism genuinely does deliver is something else, and it is not nothing: a precise account of the combination problem — when and exactly how conscious observers merge into larger ones, and how they come apart. That is a real contribution to a real problem in philosophy of mind. It is simply a different problem from the hard one, and the paper's own list of desiderata distinguishes them.

6. The proposed empirical test

To its credit, the paper does not stop at metaphysics. The route to a test runs through a mathematical coincidence.

Physicists studying scattering — what happens when particles collide — have found that certain calculations become dramatically simpler in structures called positive geometries, which live outside spacetime rather than within it. Those geometries are classified by combinatorial objects called decorated permutations. In earlier work (Fusions of Consciousness), Hoffman and colleagues showed that the same decorated permutations classify the recurrent structures of Markov chains.

Same classification, two unrelated fields. The proposal follows: particles in spacetime are projections of recurrent structures in networks of conscious agents. If so, mass, spin, energy and momentum are projections of the corresponding chain properties.

The concrete test they name is the internal structure of the proton — whether a suitably chosen master matrix, sampled at different rates, can reproduce the measured momentum distributions of quarks and gluons across scales.

The work is under way, not merely proposed. §8 states it plainly: “Computer experiments are currently being conducted to see if these proposals allow us to reproduce the known momentum distributions of the quark-gluon innards of the proton, at various levels of spatial and temporal resolution. The computational complexity is, as might be imagined, already enormous.” No results are reported yet — but running is a different thing from designed.

And the paper states a falsification criterion explicitly, which is the difference between a research programme and a worldview. In the authors' own words:

The main goal of this stage of CA theory is to be able to identify, within CA dynamics, the decorated permutations relating to physical scattering processes… Is it possible to generate these physical decorated permutations within our Q-processes on a state space of suitably chosen size? … Note that CA theory is falsifiable: if it can be shown that any physical decorated permutation cannot be found within its Markov dynamics, the theory is falsified; at best, it is incomplete.

That is a real criterion. Certain decorated permutations are known to be physical — they describe actual gluon scattering. If those cannot be generated inside the conscious-agent dynamics, the theory fails. One can argue about how tractable the search is; one cannot say the authors declined to name a condition under which they would be wrong.

A second, softer bridge to experiment appears in §7. Since Stern-Gerlach measurements at orthogonal orientations are uncorrelated and at non-orthogonal ones correlated as a function of angle, the authors suggest the trace logic supplies a candidate correlation measure: take two kernels with a join and a meet, evaluate the stationary measure of the join on the states of the meet, and the result runs from 0 (minimal correlation) to 1 (maximal). Offered as “a possible measure” and not developed — but it is the one place the meet-and-join machinery is pointed at a laboratory.

The candour elsewhere is intact. On the proton work: “It may turn out that the master matrix only needs to have three sets of nodes that are strongly bound rather than completely confined. We shall see. That is one of the points of doing a computational experiment.”

7. Where the trilogy agrees — and where it does not

The agreement is foundational, and this site has been built on a good deal of it. Consciousness is fundamental rather than produced by matter. Spacetime is not the bottom layer; it is an interface, a headset, a render. Anyone who thinks the receiver model is fringe should sit with the fact that a UC Irvine cognitive scientist, a mathematician, and a physicist are developing it with this much technical apparatus. See the companion page on Hoffman's interface theory.

The disagreement is specific and it is worth stating without hedging, because this site cites Hoffman approvingly in several places and a reader could reasonably assume the positions coincide.

The paper's claim: “CAT says consciousness has no physical substrate… spacetime is doomed, and with it any physical substrates inside spacetime. Requiring such substrates is an anachronism.”

On that view, the entire substrate question — the question Why biology? is built around, and which the trilogy treats as the decisive question of the coming century — is a category error.

The trilogy's position is narrower, and deliberately so. Grant that consciousness is fundamental. Grant that spacetime is an interface. What remains is the question of what happens inside the interface — and inside it, on this framework's reading, biology has primacy. Unless and until shown otherwise, living substrate is the exclusive channel for the transfer and development of consciousness out of the fundamental field into a localised receiver. Not because matter generates mind. Because the coupling appears to require what only living systems have so far done: the self-produced boundary, the metabolic cost, the bioelectric field, the finitude.

These claims are not strictly contradictory — they answer different questions. Hoffman describes the ontology outside the headset, where substrate is not a category that applies. The trilogy describes what the headset requires in order to render a subject at all. But they pull opposite ways in practice, and the trilogy commits to something his framework does not: that within the rendered world, which substrate you are makes a decisive difference to what reaches you.

One methodological observation, offered without hostility. Hoffman's conscious agents are substrate-free by construction. They are Markov kernels — mathematical objects with no boundary to maintain and no metabolic cost to pay for persisting. That is a legitimate modelling choice and it buys the theory real power. But a formalism's silence about substrate is not evidence that substrate is irrelevant; it is a consequence of what the formalism was built to abstract away. The trilogy's wager is that the thing abstracted away is the thing doing the work.

The disagreement is meant to be settleable. If receiver-signatures appear in a substrate with no autopoietic organisation, the primacy claimed here is wrong and Hoffman's indifference to substrate is vindicated. That is a better thing to have than an agreement papered over.

Two idealisms: why Kastrup has a gate and Hoffman does not

The obvious objection to everything above is that it looks like a fight the trilogy cannot win. If consciousness is fundamental, how could substrate possibly matter? Surely holding both is incoherent, and the trilogy is trying to have idealism while smuggling in a materialist criterion.

It is not incoherent, and the cleanest demonstration is that another idealist — working from the same premise as Hoffman and arriving at the opposite conclusion — has already built the machinery. Bernardo Kastrup is a consciousness-fundamental idealist who denies flatly that rocks or computers are conscious. Seeing how he does it shows exactly where the fork is.

Kastrup's move is dissociation. On his analytic idealism there is one consciousness — mind-at-large — which is experiential but instinctive rather than self-reflective. Individual subjects are not primitives. They are carved out of the universal subject by dissociative boundaries, on the model of dissociative identity disorder, where a single psyche produces alters that cannot access one another's inner lives. You and I are alters.

Then the identification that does the work: matter is what mental process looks like from outside a dissociative boundary, and life is what dissociation itself looks like from outside. Metabolism is not evidence that something is an alter. Metabolism is the image of one.

So the rock resolves through a distinction Kastrup insists on: constitution versus individuation. A rock is constituted of consciousness — everything is; there is nothing else for anything to be made of. But it is not a consciousness, because no dissociative boundary carves out a rock-subject. There is no what-it-is-like-to-be-a-rock, not because rocks are made of inert stuff, but because nothing there is producing a private interior. The rock is part of what mind-at-large experiences, not something that experiences.

Silicon receives the same verdict and Kastrup is blunt about it: a computer is a rock with complicated behaviour. No metabolism, no self-produced boundary, no dissociation, therefore no interiority — however sophisticated the outputs. He adds a point worth keeping: a computer's boundaries are observer-relative anyway. Where does the machine end — at the case, the power supply, the network? Dissociation is meant to be a natural fact, not a question of where an engineer draws a box.

The fork, stated precisely. Hoffman is a pluralist: conscious agents are the primitives, there are unboundedly many, and they combine and decompose. With subjects at the bottom of the ontology, no gate is available — everything whatsoever is a projection of agents, so there is no principled line between what has an interior and what does not. Kastrup is a monist with a boundary criterion: one consciousness, and subjects are derivative, produced by dissociation. Precisely because subjects are derivative, there can be a fact of the matter about where they occur and where they do not.

That is the whole difference. Two thinkers, the same starting premise, opposite verdicts on machine consciousness — and the divergence traces to a single structural choice about whether subjects are basic or made.

Which makes Kastrup the ally in this dispute that Hoffman is not: consciousness fundamental and a principled substrate gate. The autopoiesis essay leans on this, and on the striking fact that Kastrup and Michael Levin — an idealist philosopher and a working developmental biologist, starting from explicit metaphysical opposites — converge on metabolism and autopoiesis as necessary conditions.

Two honest complications, because the alliance is not free.

Kastrup's threshold is itself undefended. He draws the line at metabolism but does not say how much, or what kind. A bacterium metabolises. A slime mould metabolises and solves mazes with no nervous system. Pressed, he restricts the claim to biological organisms broadly rather than offering a principled boundary inside biology — which is exactly the gap Why biology? §5 tries to fill by adding Levin's bioelectric gradient underneath the gate.

And Kastrup's gate is not a receiver model. This matters more than it first appears. On his account biology does not receive consciousness across a boundary — biology is what a dissociation looks like from outside. The antenna vocabulary this site uses implies two things, a transmitter and a receiver, and Kastrup would say that quietly reinstates the dualism idealism was supposed to dissolve. For him there is one thing under two perspectives: first-person is experience, third-person is matter, and no transmission occurs between them because there are not two places.

The trilogy's answer is that “receiver” is phenomenological shorthand — a description of how the coupling presents clinically, from inside a substrate that is manifestly not producing what it undergoes — rather than a commitment to two ontological substances. That answer is available and this framework takes it. But it should be said out loud rather than left to a metaphor to carry, because the metaphor on its own does imply a transmitter, and a careful idealist reader will notice.

A third position: Emilsson's topological pocket

Kastrup and Hoffman are not the only two ways to run this. Andrés Gómez Emilsson, of the Qualia Research Institute, occupies a genuinely distinct third position — and it happens to answer the rock question with more precision than either.

He calls the view physicalist idealism. The substrate of reality is conscious, but not in the full sense of a world-simulation with a self in it. What is everywhere is something rawer: in his words, “the raw precursors to qualia… proto-qualia… qualia of a more basic kind.”

Then the part that matters here. Asked whether that makes a rock conscious, his answer is no, and the reason is specific:

I don't mean to say that a rock is made of specks of blue, because even blue might require a little bit of this perspectival construction, or phenomenal space… so you may require some further conditions, some symmetry breaking, some observer embedded within the topological pocket that instantiates that experience.

So: raw proto-qualia are ubiquitous, but a particular experience — blue, this moment, a point of view — requires a bounded region with the right internal structure. The boundary is topological: a pocket closed off by the geometry of a physical field, not by a psychological dissociation and not by a statistical screen.

Where that leaves the three of them. Emilsson is nearer Hoffman on constitution — consciousness really is everywhere, in some minimal form — and nearer Kastrup on individuation, because there is a real fact about where subjects begin and end. He gets a gate without paying Kastrup's price of a single universal mind, and he keeps ubiquity without paying Hoffman's price of no gate at all.

The part most useful to this site is not the boundary claim but the causal one. Every receiver-model account eventually has to answer the question: what does consciousness actually do? If it does nothing, it is epiphenomenal and evolution had no reason to build it.

Emilsson's proposal is that unified experience is a hardware accelerator. The sensory fields have different dimensionalities — touch roughly three-dimensional, vision roughly two, hearing roughly one — and something has to let them talk to each other fast. A holographic central core, where those streams meet, is what performs that translation. Consciousness, on this account, is not a by-product of the computation; it is the mechanism that makes a certain kind of cross-dimensional integration cheap enough to be worth having.

His supporting observation is that vision and touch are not really separate to begin with: visual experience carries tactile counterfactuals inside it — if I reach there, I will feel warmth — and that ordinary experience contains small phase lags between the sensory fields which themselves encode information (how far away a thing is, how many steps to reach it). He points to reports from deep meditative cessation, where practitioners describe the fields collapsing into phase alignment, a wave in the tactile field becoming indistinguishable from the same wave in the visual field. Which is worth noting alongside the meditation essay, though it rests on first-person interview reports and should be weighted accordingly.

And an argument against digital consciousness that arrives from an unexpected direction. Emilsson notes that if you simulate a system with genuine physical speedups on a classical computer, you do not get the speedup — you get the opposite. You must slow the whole simulation down to keep appearances consistent while you compute the expensive part. From inside the simulated world the object still looks fast; from outside, you have paid more, not less.

Apply that to the hologram and the conclusion follows without needing any premise about metabolism or life: a digital system cannot recruit the acceleration, only simulate it at a cost. Whatever consciousness is contributing, on this account, is precisely the thing a classical substrate cannot get for free. That is an independent route to the conclusion Why biology? argues on entirely different grounds — which is worth more than an ally who already shares the premises.

Caveats, and they are substantial. The Qualia Research Institute works outside mainstream academia and this material comes from a recorded talk rather than a paper. Emilsson is himself explicit that it is speculative — “I'm going to hypothesize,” “my money is on” — and the research programme he describes is proposed, not run: there are no results, only a design. The dimensional claims about the sensory fields are impressionistic rather than measured. His criticism of Integrated Information Theory he flags as controversial himself. And one difference from this framework should be named: Emilsson is a physicalist idealist who wants the hologram implemented in actual physics — nonlinear optics, field topology in real brain tissue. That supports the biology-primacy claim strongly, but his field is the electromagnetic one, not a consciousness field standing behind physics. The agreement about substrate is real; the ontology underneath it is not the same.

→ The talk is on Watch & Listen.

8. A useful contrast: Markov kernels versus Markov blankets

Readers who have encountered Karl Friston's work will recognise the word “Markov” and may assume the two frameworks are talking about the same object. They are not, and the difference turns out to matter here.

A Markov kernel is dynamics — the rule for how a system moves from state to state, as described in §1.

A Markov blanket is a boundary. It is a set of variables that statistically screens off an inside from an outside: given the blanket, internal and external states are independent. Friston splits it into sensory states (world influencing interior) and active states (interior influencing world). On the free-energy principle, anything that persists as a distinct thing must behave as though it were minimising surprise about its sensory states — otherwise it dissolves into its surroundings.

The contrast is exactly the disagreement in §7, expressed mathematically. Hoffman's conscious agents have kernels but no blanket: nothing to maintain, no boundary to defend, no cost paid for continuing to exist. A living system is defined by the opposite — it is the thing that spends energy holding its boundary against dissolution, continuously, or stops being a thing at all.

Which is to say: the same mathematical vocabulary that lets Hoffman abstract substrate away can be used to say precisely what substrate does. See Why biology? §4 for that argument developed properly, with the objections to Friston's framework named alongside it.

What the authors themselves flag as unfinished

Worth listing, because a fair reading requires it and because the authors are more candid than their popularisers.

The theory of observation is an idealisation. Computing a trace requires infinite sampling. Real observation is finite, and the paper's own fourteen-step example is, in their words, “still far from the correct matrix.”

The join is not fully solved, and the authors say so twice. Working out when two observers have a least common superior, the appendix reaches a case and prints, in the published text, “If l = 1 the meet does not exist???” — three question marks, left in. A remark shortly after is equally candid: “We have not here explored general solutions. It is an open question whether solutions always exist and, if so, are unique.” A special case is solved cleanly; the general case is open. This is a load-bearing gap, since the combination problem is what the formalism is being advertised to handle, and one should read the combination claims with it in view.

The Lebesgue-order results are imported, not proved here. Appendix A.5 summarises definitions and propositions from a prior paper and says explicitly that the proofs live there. So half of the observation-to-belief homomorphism rests on work outside this document.

Two central proposals are labelled conjectures — that trace order tracks entropy rate, and the claim about minimal commute times, supported by a single two-dimensional plot rather than a proof.

A structural gap in the particle correspondence. Two separate recurrent classes cannot interact at all in the formalism, which means the scheme as it stands cannot describe particles separated in space or time — that is, nearly all of them. The authors state this plainly and say the proposal needs generalising.

The speculative extensions are marked as such. Black holes as an overflow of the “spacetime headset's” channel capacity; relativistic reference frames emerging from traces. Both are floated in the conditional and neither is developed.

One further item deserves flagging, because it is load-bearing and receives two sentences: the authors note that the trace operation “violates an assumption similar to ‘statistical independence’… meaning that the probabilities (after tracing) are not independent of the tracing parameter choices.” That is, in effect, a violation of measurement independence — the assumption Bell's theorem needs. It is a significant claim, and it is left undeveloped. See the Bell explainer for why that assumption carries so much weight.

The short version

A serious, technically ambitious attempt to define what an observer is, starting from consciousness rather than matter. Its central idea — that observing something means being a compressed version of it, so that no observer stands outside what it observes — is genuinely novel and genuinely interesting, and it is proven to be mathematically well-behaved.

Everything downstream of that — mass, spin, particles, the route to experiment — is proposal rather than result, and the authors mostly say so. It does not solve the hard problem; it relocates it, and it is candid that its own foundation is unexplained. What it does contribute is a precise treatment of how conscious subjects combine and dissociate.

It is worth taking seriously. It is not yet worth taking as settled. And on the one question this site cares about most — whether substrate matters — it points firmly in the opposite direction from the trilogy, which is exactly why it belongs here rather than being quietly left out.

Two things raise it above most work in this territory. It names a condition under which it would be wrong, and computer experiments are already running against it. And it declines the hard problem openly rather than claiming a solution it does not have — a theory from consciousness, not of it. Whatever one concludes about conscious agents, that is the right way to hold a position this speculative, and it is the standard this site tries to hold itself to.

9. The follow-up — Traces of the Other

Section 6 left the DMT question where the evidence left it: what the conscious-agent reading of psychedelic entity encounters lacked was a prediction distinguishing a channel was opened from a model was perturbed. Both accounts predict vivid, structured, cross-subject-consistent experience.

That challenge has since been taken up directly. Andrew Gallimore, Niffe Hermansson and Donald Hoffman have a follow-up preprint — Traces of the Other – Are DMT Entities Real? DMT Phenomenology in the Framework of Conscious Realism ↗ (2026, PsyArXiv) — and it is an attempt to make the question answerable rather than to answer it. What follows is from the paper itself.

The proposal

Evolution has concentrated our experience-dynamics inside a basin of attraction the authors call the Consensus Reality Space — the region of experience space the qualia kernel keeps returning to. It limits which experiences are reachable and, critically, which influences from other agents can register at all. The hypothesis is that DMT perturbs the interface hard enough to push experience outside that basin, into regions where the evolved constraints no longer hold:

Our working hypothesis is that some DMT entities might correspond to traces of conscious agents with Markovian dynamics that are, compared to consensus reality, highly unusual or exotic and, as such, are normally imperceptible but which can be perceived when DMT alters the structure and dynamics of our interface.

They are careful that this is not a claim about seeing reality more truly — “this should not be understood as implying that there exists a more complete or 'truer' form of experience prior to this filtering” — and the expansion itself is argued by analogy (phase transitions, molecular vibrational modes, simulated annealing) rather than derived.

“Trace” here means something different from §2

This matters for anyone reading the two papers together. The word is technical in both, but the construction is not the same.

In Traces of Consciousness, A is a trace of B: A is a compression of B's dynamics. In Traces of the Other, a trace is a subset of the observer's own experience space — the states reachable under the channel carrying another agent's influence, with the transition structure induced on them. It is the trace of B's influence inside A, not A as a compression of B. Same word, opposite orientation.

The authors are explicit about how little it licenses: “There is never any direct mapping between the structure and dynamics of an agent and how its trace is represented in the interface of another agent.” The same agent may register as agentic to one observer and as incoherent noise to another.

Two corrections to how this paper is described elsewhere. It does not use the trace-order machinery of the paper above — the construction is defined from scratch with no citation, and Traces of Consciousness is cited only twice, for unrelated points. (The collaboration's own page ↗ says the Institute “will apply the trace logic for conscious observers and the full mathematical apparatus of conscious agent theory” — future tense. That is the programme's ambition, not what this first paper does, and the gap between the two is probably why the connection gets asserted more strongly elsewhere than the text supports.) And it never mentions Levinas or Derrida. There is no acknowledged philosophical genealogy for the title; the likely proximate source is Michael, Luke & Robinson's 2021 field study An Encounter With the Other, cited twice in the paper. Readers who hear Levinas's trace of the Other — the mark of what was never present as an object, which nonetheless obligates — are hearing a real structural echo, but it is the reader's, not the authors'.

The phenomenological case rests on one disanalogy

Entity encounters are reported in about 45% of DMT experiences across an analysis of 3,778 trip reports, and in 72% of subjects in one naturalistic study. The load-bearing statistic, though, is an absence: “humans – by far the most commonly encountered organism in the natural waking environment – are only rarely encountered in the DMT state (<5%).”

The argument: dreams and psychosis also involve loosened top-down control, and both are populated by ordinary people. If DMT were simply unconstrained endogenous generation, it should default to familiar content. It does not.

Four experiments, deliberately complementary

All rest on extended-state DMT (DMTx) — target-controlled infusion holding a stable state for up to 120 minutes, already published and shown tolerable.

1. Hidden external variable. A computer in a sealed room switches randomly between a blue and a yellow screen; the test is whether an entity can express a modulation tracking it.
2. Dynamical-capacity probes. Not intelligence tests — and the reasoning is the sharpest thing in the paper: “the relevant question is not 'Are you smart?', but 'What kinds of dynamical structures can you stably instantiate and control?'” Can an entity colour an arbitrarily complex non-Euclidean surface with no two adjacent regions matching; stay coherent through several simultaneous independent transformations; resume an interrupted construction with no cue?
3. Multi-subject convergence under isolation, testing for non-trivial agreement on uncued detail.
4. Information deposit and retrieval. One subject gives an entity a random word or number; a second subject attempts to retrieve it.

The limitations are the authors' own, not extracted from them: the hidden-variable test “requires the active cooperation of the entity, which is obviously not guaranteed”; the capacity probes depend on subjective report to score. Nothing has been run.

The concession is larger than most summaries convey

On the predictive-processing account — that DMT suppresses high-level priors and lets pattern detectors impute agency to unconstrained bottom-up signalling — they write:

whilst the standard hallucination explanation must remain as the default, most parsimonious explanation for DMT's effects, considering and empirically testing alternative hypotheses… is both warranted and scientifically tractable.

They concede their own alignments are “suggestive rather than conclusive,” and note that within conscious realism hallucinations are themselves agent interactions — which blunts the dichotomy the title trades on considerably. The closing sentence is the honest one: “our purpose here is not to claim that DMT entities represent external conscious agents, but to show that, given a formal theory in which consciousness is fundamental, such a possibility can be coherently formulated and experimentally investigated.”

On sources. The formal apparatus is the authors' own and the DMTx work is published. But the qualitative phenomenology carrying much of the argument comes from two Reddit posts, an Erowid report, and blog-and-video material from Andrés Gómez Emilsson — whose talk is elsewhere on this site, with its caveats noted there. The abstract promises “testable predictions”; the body delivers retrospective alignments plus four experimental designs. That is a research proposal, not a result, and the authors do not pretend otherwise.

Gallimore, A. R., Hermansson, N., & Hoffman, D. D. (2026). Traces of the Other – Are DMT Entities Real? PsyArXiv preprint. Read the paper (free full text) ↗ — not peer-reviewed. Gallimore is at the Okinawa Institute of Science and Technology; Hermansson and Hoffman at the Trace Institute, which takes its name from the trace-chain result walked through above. The work is a joint programme between the Institute and Noonautics — Trace × Andrew Gallimore ↗ — with Noonautics running the extended-state DMT protocol and the Institute supplying the formalism. A recorded public conversation between Hoffman and Gallimore accompanies it.

Donald Hoffman, Chetan Prakash & Swapan Chattopadhyay, “Traces of Consciousness,” Preprints.org, posted 17 October 2024, doi:10.20944/preprints202410.1305.v1. Read the paper ↗ — free full text, not peer-reviewed. Related on this site: Hoffman & the interface theory, Why biology? The autopoiesis test for receivership, Chalmers and the hard problem, Bell's theorem, and the Gallimore-on-Hoffman conversation on Watch & Listen.