What Lies Beneath Spacetime?

What Lies Beneath Spacetime? Donald Hoffman, Recursive Trace Logic, and the Mathematics of the Observer

Nicole Flynn, Symfield PBC, September 15, 2026

Donald Hoffman’s physics claims remain conjectures. That is precisely why his program is worth watching. He may be building the mathematics that forces science to reconsider what it has mistaken for reality. At present, that possibility remains open and deserves serious attention. Hoffman is best known publicly for The Case Against Reality, published in 2019. The book argues that evolution did not shape human perception to reveal the objective world. It shaped perception to guide adaptive behavior. What we experience as objects in space and time may therefore be less like a window onto reality than a desktop interface: useful, reliable and systematically connected to something real, but not a literal depiction of what operates underneath.

In The Case Against Reality, Hoffman introduces the analogy that has come to define his public argument: perception functions like a computer interface. The icon on a computer desktop does not resemble the transistors, voltages, or code that sustain it. Yet it is not arbitrary. It compresses hidden complexity into a form through which a user can act. Hoffman proposes that physical objects, and perhaps spacetime itself, function in much the same way.

That argument is provocative, but it is no longer the most consequential part of his program. Hoffman and his collaborators are now working on the formal machinery beneath it: conscious-agent theory, trace logic and, most recently, Recursive Trace Logic. Their ambition is considerable. They are attempting to derive the structures of spacetime, quantum theory and particle physics from a mathematical theory in which observers, rather than physical objects, are fundamental. This is not merely another philosophical declaration that consciousness is mysterious or that reality is an illusion. Hoffman is proposing mathematics. The mathematics begins with Markov processes.

The observer as a Markov process

In the conscious-agent framework, an observer is represented by a measurable space of possible experiences and probabilistic kernels governing transitions among perception, decision and action. The construction is deliberately minimal. It does not begin with neurons, bodies or objects located in a pre-existing spacetime. Those belong to the interface the theory ultimately hopes to recover. The central mathematical development is the trace chain. Suppose a larger Markov process has transition kernel P, but an observer has access only to a subset A of its states. The dynamics available to that observer are not obtained simply by cutting out the A × A portion of the transition matrix. The process may leave the visible subset, pass through one or many hidden states and eventually return.

The effective transition kernel on the observer’s visible states is

Each term represents a possible excursion through the inaccessible complement A′ before the process returns to what the observer can experience. The hidden states are not simply discarded. Their effects remain present in the probabilities of the visible process. This is an elegant and important idea. An observer does not receive a damaged copy of a complete world. The observer encounters an effective world whose apparent dynamics have been shaped by processes that remain outside its window. Different windows into the same larger process can therefore produce different experienced worlds, each internally coherent and none equivalent to the whole.

The Trace Chain Theorem gives this intuition formal expression: for a Markov kernel and an appropriate visible subset, there is a unique effective trace chain describing the dynamics restricted to that subset. From relations among such traces, the researchers construct a partial ordering of observers and a corresponding logic that is locally Boolean but not necessarily Boolean globally. This is where Hoffman’s headset metaphor stops being merely illustrative. It becomes a mathematical account of how inaccessible structure can remain causally legible inside a restricted interface without becoming directly visible within it.

Why the trace becomes recursive

Ordinary trace logic considers an observer-window on a larger process. As described in the Trace Institute’s current research program, Recursive Trace Logic extends the construction through a hierarchy of Markov kernels: a policy level governing state-space transitions, a meta-policy level, and beyond. The observer’s world is therefore not confined to a fixed inventory of states with fixed transition probabilities. The conditions under which states and transitions become available may themselves evolve. This matters because a theory capable of generating spacetime cannot quietly assume spacetime in its foundations. Nor can it merely place familiar physical objects inside a larger invisible container. The mathematics must allow the structures through which a world appears, its effective states, relations, scales and regularities, to arise from dynamics that do not already presuppose them.

Hoffman’s program is attempting exactly this escape from an inherited box. That willingness should not be underestimated. Foundational physics has repeatedly discovered that concepts once treated as the stage of reality were contingent features of a deeper description. Absolute space yielded to spacetime. Classical trajectories yielded to quantum states. Local particles have increasingly yielded, in some calculations, to combinatorial and geometric structures that do not treat spacetime locality as fundamental. The question is not whether we are permitted to reconsider spacetime. We are. The question is whether the proposed replacement can recover what spacetime and quantum field theory already explain while also showing why those structures appear. I hope it can. I already suspect that spacetime is inherited descriptive scaffolding rather than fundamental reality; if Hoffman’s mathematics exposes what generates it, even if his final ontology differs from mine, we all gain from the result.

The unreleased spacetime mathematics

The Trace Institute’s published conjectures describe new work in which Minkowski spacetime is proposed to emerge as a limiting behavior of Markov chains associated with n-cycles as n tends toward infinity. The broader research program proposes that curved spacetime arises from suitable non-cyclic chains, that cosmological behavior appears in long samples of trace processes and that the failure of spacetime at the Planck scale follows from properties of the underlying trace structure.

The detailed spacetime derivation has not yet been publicly released. I want the definitions, the limiting procedure, and the correspondence between formal parameters and physical observables. Until those are in hand, we cannot responsibly say what has been established. The claim is specific enough, and consequential enough, to deserve anticipation rather than dismissal.

The decisive question will be what, precisely, emerges. A compelling derivation must do more than produce a geometric object that can be interpreted as spacetime after the fact. It must explain the appearance of Lorentzian signature, invariant interval, causal structure, relativistic transformations and dimensionality from the stated Markovian premises. If the mathematics genuinely produces those structures without inserting their equivalents in advance, Hoffman will have done something far more important than supply another metaphor for emergence.

He may have identified a generative route from non-spatiotemporal dynamics to the structure physics has treated as its arena. I want to see that. Not because I have decided in advance that it succeeds, and not because I want it to fail. I want to see the definitions, the limiting procedure and the correspondence between formal parameters and physical observables.

From Markov chains toward quantum field theory

The proposed bridge to quantum field theory is equally striking. Hoffman’s collaborators map recurrent communicating classes of Markov chains to decorated permutations. Decorated permutations already organize cells of the positive Grassmannian, a mathematical structure important in modern work on scattering amplitudes. The program also connects Markov polytopes with positive geometries such as associahedra, which can encode particle-scattering relationships without beginning from the conventional picture of particles traveling locally through fundamental spacetime.

The intended chain of construction is roughly

This is not an arbitrary sequence of analogies. There are real mathematical correspondences at its intermediate stages. The research program aims further toward free-particle wavefunctions, the Born rule, elementary-particle classifications, entanglement and eventually the structures of the Standard Model. Those aims should presently be described as aims. The public materials identify eight physics conjectures and they do not yet constitute a published derivation of quantum field theory in full. That boundary does not diminish the work. It tells us where the work actually stands.

If one underlying formalism were to recover spacetime, quantum probabilities, scattering structure and particle classes, not by separately fitting each result, but as consequences of the same generative architecture, that convergence would demand attention. The strongest evidence for the framework would then be neither its philosophical beauty nor its use of the word consciousness. It would be the amount of established physics it recovers, the assumptions required to recover it and any novel prediction that distinguishes the theory from its competitors. That would be an extraordinary gain for everyone.

What mathematics can prove about consciousness

Hoffman takes consciousness as fundamental and is building the mathematics that would follow from that premise. I hope more of that construction reaches the public record soon. A careful distinction is still necessary, not to weaken the project, but to understand what would count as its success. Mathematics can prove what follows from specified primitives, definitions and axioms. If an entity is defined as a conscious agent with a measurable experiential space and Markovian dynamics, one can prove theorems about its traces, compositions and asymptotic behavior. Mathematical consistency alone, however, does not establish that the formal entity possesses phenomenal experience or that reality is ultimately composed of such entities.

That does not make the conscious-agent premise illegitimate. Every new theoretical program begins by risking a premise that has not yet earned general acceptance. One must first suppose that something can be built before undertaking the work of building it. A conjecture is not an epistemic failure. It is permission to construct.

The issue is what the construction returns. If consciousness is assumed fundamental, conscious agents are defined mathematically, and the formalism is shown to be internally coherent, the result establishes the consequences of the assumption. If the same formalism then produced the unexpected structures of relativity and quantum theory, explained why they appear, survived comparison with observation, and made successful predictions unavailable to rival frameworks, the achievement would be extraordinary. The premise would have acquired genuine explanatory authority, not because it had been assumed, but because of what the resulting architecture could uniquely recover and predict. Not certainty. Authority.

There is a profound difference between declaring that a premise has been proved because its mathematics is coherent and watching a premise earn its place by the reach, precision and vulnerability of what it generates. Hoffman is disciplined enough to understand that distinction. His confidence appears to function not as a substitute for the work, but as the condition under which he is willing to attempt it.

The boundary of every formal world

My own work approaches several of these questions from a different direction, although I have long found Markov mathematics congenial. I share Hoffman’s suspicion that spacetime may be an interface rather than the substrate of reality. I also share his insistence that an observer cannot be treated indefinitely as an undefined spectator positioned outside the theory.

I should be explicit about my position here: I am not a mathematician. I develop structural frameworks, follow the mathematics that appears relevant to them and consult trained mathematicians and scientists to determine where an intuition survives formalization. I particularly like Markov mathematics because it gives disciplined expression to transitions, conditional futures and the boundary between what a present state captures and what it leaves unresolved.

That interest has already led me toward a neighboring construction in my own unfinished work (SNS–Tiny Delta Correspondence v0.1, July 16, 2026). The earlier correspondence represented a realization by the tuple

Here M is the interaction domain; E is the internal-state bundle; Φ is the current state; h is accumulated history; Π maps internal state into realized transport; 𝓓_μ is the operator family; W is a witness rule; and 𝓐ₜ is the current admissible configuration set within the ambient state space 𝓧 = Γ(E). Its schematic evolution was

Here F_μ is the candidate generator and Pₜ is not necessarily an ordinary linear projector. It is a local routing map into T_{𝓐ₜ}(Φₜ), the admissible tangent cone at the current state, rather than necessarily the tangent space of a smooth manifold. A finite witnessed event enters as consequential history only when it leaves a persistent distinction and changes the future generator, admissible set or probabilities of later transitions. The existing formulation therefore already asks whether realized history can alter not simply the next state, but the geometry of what may occur next.

The Mori–Zwanzig projection formalism makes this question exact. Choose a projection 𝓟 onto functions of the variables one has decided to keep — here the realized variables Yₜ = (Φₜ, hₜ, 𝓐ₜ, Lₜ) — and let 𝓟⊥ = 1 − 𝓟. For any dynamics with generator 𝓛, the evolution of the kept variables obeys

The first term is the Markovian law on the kept variables; the memory kernel K and the noise Fₜ are generated entirely by the dynamics that were projected out. In Hoffman’s trace construction the projected-out dynamics are the excursions through A′, and the kernel is what the trace chain resumes into P_A. In the present proposal the projected-out dynamics are the residue Dₜ. The question “does the realized path close?” is the question “does K vanish when 𝓟 projects onto realized variables?” My further question is whether the kernel’s form — not only its presence — is conditioned by the organization that realized history has already produced: whether K depends on 𝓐ₜ, so that prior organization alters how history bears on what may happen next.

If history changes the admissible geometry of future motion rather than merely determining the next state, then a mathematical account of consciousness would require more than recursion over a fixed state space. It would require accumulation, load, branching, path-dependent admissibility and sequencing that is not fully controlled by the observer. An organism does not determine the order, simultaneity, intensity or duration of every event that reaches it, nor does it independently choose the apparatus through which an event is registered. The resulting sequence is neither wholly random nor centrally controlled. It is partially endogenous, partially imposed and conditioned by the structure through which it becomes available. Conscious dynamics, on this proposal, move through a branching landscape that realized history partly constructs.

The following coupled system is a proposed representation of that architecture, not a completed derivation (Flynn, proposed formulation, September 15, 2026):

The coupled system is a time-stepped realization of the continuous flow above. It is presented as a schema: its spaces, maps, measurability conditions and contingency law must be supplied for any specific realization. Because F_μ depends explicitly on t, such a realization would be time-inhomogeneous unless time were included in the state. Here W is a fixed, realization-specific witness rule. The contingency term ξₜ is either supplied by a stated exogenous law or assigned a conditional law determined by the current augmented state.

The selection map partitions the prospectively generated candidate set 𝓑ₜ into the realized transition rₜ and the unrealized set Uₜ. Nothing in this formulation declares Uₜ nonexistent or simply discards it. The residue map 𝓠 asks what, if anything, is retained, weighted, transformed, dissipated or recompressed from those unrealized candidates into Dₜ₊₁. The operational burden lies in Br: candidate transitions must be specified prospectively, independently of later outcomes. They need not include every mathematically admissible possibility; a complete realization must state what makes a transition available and what degree of activation, preparation or load permits it to contribute to residue. If Br merely returns every admissible transition, then Uₜ may be reconstructible from the admissible set and realized transition, leaving Dₜ with no demonstrated independent predictive content.

Under the displayed recursion, Dₜ is a functional of the initial residue, prospectively generated unrealized candidates, realized transitions, load and contingency history.

The realized/unrealized distinction is essential. Selection produces a realized branch, but does not necessarily render every unrealized branch dynamically null. A possibility that was available, approached, suppressed, interrupted or excluded may leave residual structure. This does not assert that every unrealized possibility remains equally present, nor that unrealized worlds persist as literal parallel realities. It asserts the narrower proposition that the history of non-realization may remain consequential, and that different unrealized branches may leave unequal residue.

The present witness may also fail to register that residue. For an incomplete or non-injective witness rule, it may be possible that

Failure of present detection therefore need not establish absence. The expression does not establish that debris exists.

The null model is 𝓖_D ≡ 0. The notation permits debris to influence later admissibility through 𝓖_D; it does not demonstrate that this channel is nonzero, necessary or distinguishable from an ordinary latent variable. The decisive test is whether changing the prospectively specified candidate set, while holding the realized transition and realized-path conditions fixed, changes later behavior or admissibility relative to that null model.

As displayed, the coupled system is a schema rather than a completed stochastic process. Once its state spaces, update maps, measurability conditions and contingency law are supplied, it may be represented as a time-inhomogeneous Markov process on the augmented state. The open question is therefore not whether memory can be placed inside a larger state, but what that state must contain. The present proposal asks whether the realized path is sufficient, or whether structured residue associated with prospectively specified but unrealized transitions carries additional predictive information. Its presence in the equations states that hypothesis; it does not demonstrate it. Hoffman’s traces and the construction below share a formal concern and do not share an ontology. Trace logic keeps the effects of states the process actually traverses but an observer cannot see. The present schema asks whether transitions that were available and not taken can leave structured residue. Hidden traversal is not unrealized possibility. Their relationship is worth investigating. Their equivalence is not assumed.

A problem you can solve this afternoon

A schema invites nothing. Here is the smallest instance I can construct, offered so that the hypothesis becomes a problem rather than a gesture. Load is held constant and suppressed.

Let the state space be 𝓧 = {1,2,3,4} and let transitions be the twelve directed edges e = (i, j), i ≠ j. The admissible set 𝓐ₜ is a subset of those edges, with 𝓐₀ all twelve. At each step, contingency assigns every admissible edge out of the current state an activation aₜ(e) ~ Uniform(0,1), independently. Then:

with θ = 1/3, λ = 1/2, τ = 7/10, and 𝓑ₜ is the single admissible edge of highest activation if no edge clears θ.

Two runs from Φ₀ = 1. In run (1) the activations are a₀(1,2) = 0.9, a₀(1,3) = 0.8, a₀(1,4) = 0.1; the candidates are {(1,2), (1,3)}, the realized edge is (1,2), and D₁(1,3) = 0.8 > τ, so (1,3) leaves the admissible set. In run (2) the activations are 0.9, 0.2, 0.1; the only candidate is (1,2), the realized edge is the same, and D₁ = 0. The two runs have identical realized paths and identical realized states. They differ only in what was available and not taken.

Under the null model the two runs are indistinguishable in law forever. Under the debris model they are not: whenever the process returns to state 1, run (1) can never realize (1,3) and run (2) can. The conditional law of the future given the realized path differs, and the difference is caused by an edge that was never traversed.

Four questions, in increasing difficulty:

1. Confirm that Dₜ is not a function of the realized path (Φ₀, r₀, …, rₜ₋₁), and characterize the smallest augmented state on which the debris model is Markov. (I believe it is 𝓧 × [0,1]¹² × 2¹², or the subset of it that is reachable.)
2. Show that, under the null model, the realized path is a sufficient statistic for the future, and under the debris model it is not. Give the conditional-independence statement precisely.
3. Identifiability: given only realized paths from many runs, can the debris model be distinguished from a null model whose transition probabilities have simply been re-fit? What sample of realized paths suffices, and is the intervention on 𝓑ₜ (change the available-but-unrealized edges, hold rₜ fixed) necessary or merely convenient?
4. Replace the threshold rule with a smooth one, Pr(e ∈ 𝓐ₜ₊₁) = 1 − σ(Dₜ₊₁(e) − τ), and ask whether the Mori–Zwanzig kernel for the projection onto realized variables can be written in closed form. If it can, its dependence on 𝓐ₜ is the object I am actually asking about.

Nothing in this toy is claimed to describe consciousness. It is the minimal case in which “available but unrealized” is prospectively defined, residue is explicit, and the null model is nested. If the instance is trivial, I would like to know exactly why. If it is not, it is the first thing worth building on. The same boundary that makes Hoffman’s program interesting is the one that requires the debris schema to remain a schema: the boundary between a formalism and the reality it describes.

Every mathematics establishes conditions of admissibility before it derives an answer. This remains true when its state space is infinite, probabilistic, recursive or capable of transformation. The formalism must still specify what counts as a state, a distinction, a relation, a transition and a valid operation. These specifications do not necessarily impose a crude or finite boundary, but they do establish the grammar within which something can appear mathematically.

Trace logic handles this problem more intelligently than many frameworks because hidden states are permitted to alter visible dynamics. Yet the effective trace remains a reduction. Distinct hidden processes can, in principle, produce the same visible kernel:

The trace may preserve the influence of what lies outside an observer’s window without uniquely identifying what produced that influence. Several generative structures may remain compatible with the same experienced dynamics. This is not an objection that defeats Hoffman’s program. It is one of the questions the program makes newly precise. If perception is an interface, could even a pre-spacetime mathematics remain an interface to something it does not exhaust? When a formalism generates an answer, what alternatives, relations or generative structures remain unresolved beside it? Does Recursive Trace Logic describe the substrate, or does it give us an extraordinarily powerful new window onto the substrate? I do not presently know. I am not sure that anyone can know before the mathematics is released and tested. What matters is that the question can now be asked without retreating into vagueness. Hoffman has moved the discussion away from declarations about illusion and toward formal relations among observers, hidden processes and effective worlds. That is already an accomplishment.

The work deserves to be seen

There is a peculiar conservatism in assuming that our inherited conceptual boxes must be permanent simply because generations of excellent scientists worked within them. Respect for established physics does not require confusing its predictive structures with final ontology. Indeed, physics itself has repeatedly advanced by separating the two. Hoffman is taking seriously the possibility that spacetime is not where reality begins. He is also accepting the corresponding burden: if spacetime is an interface, then the deeper theory must recover spacetime rather than merely speak around it. The same is true of quantum field theory. The headset metaphor opens the door; mathematics must carry the argument through it.

Perhaps the unreleased derivation will expose a hidden assumption. Perhaps it will produce a beautiful but non-unique correspondence. Perhaps it will open a route no one has previously seen. Any of these outcomes would be worth knowing. The last could alter the foundations of science. For now, the most intellectually responsible position is also the most intriguing one… pay attention. I hope Hoffman releases it soon. I hope physicists and mathematicians examine it seriously when he does. I hope it succeeds, not because success would settle every question about consciousness or reality, but because foundational physics still lacks a tested account of why spacetime appears and any framework attempting to derive that stage rather than assume it deserves serious examination. My own related construction remains a working proposal, haunting me daily as I traverse it. Corrections, criticism and technical contributions are welcome, especially on how available-but-unrealized transitions can be specified prospectively, whether their proposed residue can be distinguished from an ordinary latent variable, and what evidence would falsify the debris hypothesis.

Selected sources

• Donald D. Hoffman, Chetan Prakash and Swapan Chattopadhyay, “Traces of Consciousness,” preprint, 2024/2025: https://www.preprints.org/manuscript/202410.1305
• Donald D. Hoffman, Chetan Prakash and Robert Prentner, “Fusions of Consciousness,” Entropy 25, no. 1 (2023): https://doi.org/10.3390/e25010129
• Donald D. Hoffman and Chetan Prakash, “Objects of Consciousness,” Frontiers in Psychology 5 (2014): https://doi.org/10.3389/fpsyg.2014.00577
• Trace Institute, current research program and eight physics conjectures: https://traceinstitute.org/research/
• Trace Institute, publications and working papers: https://traceinstitute.org/publications/