Constructor theory of time

Chiara Marletto; David Deutsch

REVIEW 1 major objections 1 minor 2 cited by

Constructor theory of time

T0 review · 1 major / 1 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper claims that duration and dynamics can be recovered from constructor-theoretic laws that never mention time, by treating timers as isolated substrates whose spontaneous transitions are possible or impossible tasks.

desk verdict The paper has a promising idea and a clear exposition, but its derivation of timers from the null task conflates the empty relation with the null task, and the construction presupposes the duration it claims to define. read the letter →

arxiv 2505.08692 v3 pith:BOAVLGV3 submitted 2025-05-13 quant-ph

David Deutsch , Chiara Marletto This is my paper · ORCID

classification quant-ph

keywords constructortheorytimetimersdurationdynamicsnulltasktimelessphysicspossibleandimpossibletasks

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Constructor theory expresses laws of physics as lists of which transformations can or cannot be brought about by cyclic devices, with no reference to time. The paper tries to show that such timeless laws can still give meaning to duration and dynamics. It defines a timer as an isolated substrate with three attributes—start, running, halted—that changes by itself, and it shows that two timers of equal duration halt together while unequal ones cannot, which partitions timers into duration classes. From those classes, the paper recovers differential equations by pairing each physical variable with a timer of shrinking duration and taking a limit. If this works, traditional time-dependent laws become emergent approximations of a more fundamental timeless theory.

What carries the argument

The load-bearing object is the null constructor—the device for the empty task $\{\}$—which, because it acts on no substrate, must itself be an isolated substrate with at least three attributes $0$, $R$, $1$ plus a halt flag. The composition principle makes the null task possible, while locality ensures the timer attributes remain undisturbed. The argument then uses the possibility or impossibility of the composite task $(0,0)\to(1,1)$ to define duration equivalence classes, and takes the $\Delta\lambda\to 0$ limit of the task $(v(\lambda),0)\to(v(\lambda+\Delta\lambda),1)$ to recover differential equations.

What would settle it

Find a constructor-theory-compliant model in which the composite task $(0,0)\to(1,1)$ is possible for two timers of unequal declared duration, or in which every isolated substrate has only static attributes; either result would show that duration and dynamics cannot be defined this way.

Watch

Extended reading notes

Core claim

The central claim is that constructor-theoretic theories that support timers can define duration without any time parameter. A timer is a null constructor: an isolated substrate with attributes $0$, $R$, and $1$ and a halt flag, where the spontaneous transitions $0\to R\to 1$ are possible tasks and $1$ is static. For two timers with durations $t$ and $t'$, the composite task $(0,0)\to(1,1)$ is possible if and only if $t=t'$ (equation (9)), and impossible when $t'\neq t$ (equation (8)); this equivalence class defines the duration $t$. Dynamics is recovered by taking a variable $v(\lambda)$ of an isolated substrate, pairing it with a timer of duration $\Delta\lambda$, requiring $(v(\lambda),0)\to(v(\lambda+\Delta\lambda),1)$ to be possible (equation (10)), and taking the limit $\Delta\lambda\to 0$ to obtain $\mathrm{d}v/\mathrm{d}\lambda$. The paper claims this grounds timeless approaches to time in general principles rather than in the formalisms of existing dynamics.

Load-bearing premise

The construction collapses if no isolated physical system can change its own state by itself, with no outside help; the timer is precisely such a self-changing system, which the paper calls a null constructor with attributes $0$, $R$, and $1$ and a halt flag.

Editorial extensions

If this is right

Traditional differential equations would become emergent approximations, valid only where timers of arbitrarily small duration can be approximated.
Constructor-theoretic laws would be compatible with traditionally formulated laws, resolving the paper's opening problem about time.
A constructor-theory-compliant theory that does not support timers is not ruled out; such a world would simply have no time in the traditional sense.
Timeless approaches to quantum gravity and quantum dynamics would gain a common foundation independent of their original formalisms.
Mutually isolated identical timers necessarily stay synchronized, so the traditional notion of clock synchrony follows from isolation alone.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable extension would be to check whether the limit $\Delta\lambda\to 0$ actually converges for timers with finite physical resources, or whether discrete duration is the more primitive notion.
The existence of a null constructor may imply that any universe describable by constructor theory contains self-changing isolated substrates; this could constrain which cosmological models are admissible.
The same equivalence-class construction might be applied to define other quantities, such as length or energy, if suitable 'rulers' can be defined as isolated substrates with a halt flag.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The core idea here is worth taking seriously: Deutsch and Marletto want to ground relational time in constructor theory, treating timers and duration as emergent from the possibility and impossibility of tasks. The specific formalism—null constructors, the equivalence classes in eqs. (8)–(9), and the differential-equation recovery via eq. (10)—is genuinely new to me, and the paper is clearly written and honest about being conjectural. The discussion of why staticity can't be expressed simply as a forbidden task is helpful, and the citations to Page-Wootters, Rovelli, Barbour, and others are appropriate.

But the central derivation has a problem that the stress-test note gets right. The paper claims the null task is possible by writing the serial composition {a→b}•{c→d} = {} when b∩c=∅, then invoking the composition principle. That identifies the null task with the empty relation. They're not the same. The null task, by the paper's own definition, places no constraints on input or output—it's the universal relation, trivially possible. The empty relation, by contrast, permits no transformations at all; it's unsatisfiable and therefore impossible under the paper's own accuracy/reliability definitions. So the proof doesn't go through. That matters because the subsequent claim that a null constructor must be an isolated substrate with attributes 0, R, 1 and a halt flag rests entirely on this step. Without a valid derivation or an explicit assumption, the existence of timers is not established.

There's a second, related circularity. Section 6 introduces a 1-parameter family of sets C_t, each containing all possible timers of duration t, before duration has been given a constructor-theoretic meaning. The equivalence classes are then defined using these very sets. That's presupposing what's supposed to emerge. Maybe the authors intend C_t as a primitive, but then the paper should say so plainly rather than presenting it as a derivation.

Minor point: the recovery of dynamics in Section 7 is a sketch. After the limit Δλ→0, the differential equation is just asserted to express the dynamics. That's fine as a proposal, but it's not a derivation, and no example is given in an actual constructor-theory-compliant theory. I'd also note that the paper doesn't engage with the possibility that some constructor-theory-compliant theories simply contain no timers; it says they're not ruled out but doesn't explore what that means for the recovery of time.

Who's this for? People already invested in constructor theory or in the foundations of relational time. It won't change what we measure or build, and it makes no empirical predictions. But it's a serious attempt to unify several timeless approaches under one task-based scheme, and the flaws are instructive.

My recommendation: send it to a serious referee. The argument isn't sound as written, but the questions are substantial and a good referee could push the authors to either prove or explicitly assume the null constructor's existence, and to fix the circularity. With heavy revision, it could become a valuable contribution.

Referee Report

1 major / 1 minor

Summary. The paper aims to show how constructor theory, whose laws are formulated as timeless statements about possible and impossible tasks, can give meaning to duration and dynamics by constructing a theory of clocks and timers. The central idea is to introduce the null task, argue that the composition principle forces it to be possible, and thus that a 'null constructor' exists as an isolated substrate with three attributes (0, R, 1) and a halt flag. Such a substrate is interpreted as a timer. Section 6 defines equivalence classes of timers via the possibility of the joint task (0,0)->(1,1) on composite timers, labelled by a duration t. Section 7 proposes to recover differential equations by considering a variable v(λ) of an isolated substrate and a timer of duration Δλ, then taking the limit Δλ→0. The paper claims that this provides an explanatory foundation for 'timeless' approaches to time, independent of specific physical theories.

Significance. If the central derivation were valid, the paper would constitute a significant contribution to the programme of constructor theory, offering a relational, theory-independent account of time and dynamics. The manuscript has several strengths: it states the conditions on timers explicitly in equations (8) and (9); it carefully discusses the roles of isolation, staticity, and the halt flag; and it openly acknowledges that not all constructor-theory-compliant theories need support timers (Section 8). The paper also gives a clear taxonomy of the assumptions used in existing 'timeless' approaches. However, the validity of the central construction depends on a proof in Section 5 that the null task is possible; as discussed below, that proof conflates the null task with the empty relation and is therefore not sound. Because the subsequent definitions of timers and the recovery of dynamics rest on this step, the paper's main claim is not established. The framework may still be salvageable as a set of sufficient conditions, provided the existence of timer-supporting theories is treated as an explicit assumption rather than a derived consequence.

major comments (1)

[Section 5] The remark in Section 5 that 'a null constructor C for the null task doesn't act on a substrate and must therefore be an isolated substrate itself' is not justified. Even if the null task were possible, a device could in principle perform it by doing nothing to any substrate that might be presented, without itself undergoing a spontaneous change of attributes. The assertion that the null constructor must have three attributes (0, R, 1) and a halt flag, and that it must transform itself spontaneously while remaining isolated, is a strong physical claim that goes beyond the definition of the null task. This step is essential for the interpretation of null constructors as timers, and it needs either a proof or an explicit assumption. The paper currently presents it as an implication, but it is not one.

minor comments (1)

[Throughout] The notation [C_t ⊕ C_{t'}]_t in Section 6 is not defined precisely. The text says the composite is denoted with this symbol 'when defined as another timer belonging to C_t', but the attribute assignments for the composite and the role of the halt flag are not spelled out. A formal definition would improve clarity.

Circularity Check

2 steps flagged · score 6.0 of 10

Duration is imported through the labels on C_t, and the claimed proof that the null task is possible rests on equating the null task with the empty relation.

other [Section 5, 'The null task and constructors for it']
"Consider the null task { }, which is represented by the empty set. This task has no constraints on what substrate it is to be performed, nor does it have constraints on its allowed input or required outputs. It is therefore the most lenient task. Remarkably, the composition principle implies that the null task must be possible, {} ✔. For, upon denoting via the symbol '•' the serial composition of two tasks, the null task can indeed be written as the serial composition of two possible tasks, whose inputs and outputs are distinct from each other: {a→b}•{c→d}={}, (b∩c=∅)."

The load-bearing step is the identification of the 'null task' (no constraints, most lenient) with the empty set. In the paper's own formalism, a task is an ordered pair of attributes; a task with no constraints would be a universal relation, whereas the serial composition of tasks with disjoint output and input is the empty relation, which has no satisfying input-output pairs and is unsatisfiable, not lenient. A possible task is one that approximate constructors can bring about with unbounded accuracy; an unsatisfiable relation cannot be brought about at all. Thus the asserted equality {a→b}•{c→d}={} does not express the null task, and {} ✔ is not a consequence of the composition principle.
self definitional [Section 6, 'Timers', equations (8)-(9)]
"If timers defined under constructor theory are to resemble arbitrarily closely the timers that one can describe in the traditional conception of physics, there must be a 1-parameter family of sets 𝓒_t, each containing all possible timers of duration t. ... Hence we shall denote the composite system 𝓒_t⊕𝓒_t', when defined as another timer belonging to 𝓒_t, with the symbol [𝓒_t⊕𝓒_t']_t. ... (𝟎,𝟎)→(𝟏,𝟏) on [𝓒_t⊕𝓒_t']_t ✔ if and only if t=t'."

Duration t is introduced as a label on the family 𝓒_t before any equivalence class is defined. The composite [𝓒_t⊕𝓒_t']_t is stipulated to be 'another timer belonging to 𝓒_t', and the timers in 𝓒_t are, by definition, those of duration t. Equation (9) therefore restates the labelling convention: for t'=t the composite is declared to belong to 𝓒_t, and for t'≠t it is not. It does not measure or define duration independently of the pre-labelled family. Section 7 then uses 'a timer with duration Δλ' in equation (10), so the differential equation is taken with respect to a parameter that was assumed as part of the definition of 𝓒_t. The recovery of duration and dynamics is thus conditional on an already time-labelled input.

full rationale

The paper is honest that it is giving conditions for timer-supporting theories and explicitly notes that theories not supporting timers are not ruled out. However, the derivation chain contains two load-bearing points that prevent the central claim from being fully first-principles. First, the claimed proof that the null task is possible conflates the null task with the empty relation; under the paper's own definition of a possible task as one approximable with unbounded accuracy, an unsatisfiable empty relation cannot be possible, so the existence of the null constructor is not actually derived. Second, the equivalence class of timers of a given duration is not derived from timeless task possibilities: the family 𝓒_t is introduced already labelled by duration t, and equation (9) merely encodes that labelling. Consequently, the recovery of differential equations in Section 7 differentiates with respect to a parameter that was put into the definition of 𝓒_t. These are structural reductions of the advertised result to its inputs, not merely missing details or self-citation issues. The score is 6 rather than higher because the paper carefully frames its main theorem as conditional on timer-supporting theories, and the later construction does not claim to prove that such theories exist; the circularity is partial, but it affects the central derivation of duration.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

No numbers are fitted and no empirical constants are introduced. The construction rests on the axioms listed: the composition and locality principles of constructor theory, the existence of accuracy measures, the existence of isolated timer substrates (the load-bearing assumption), and differentiability for the limiting procedure. The only invented object is the null constructor or timer, an idealization with no independent empirical handle.

assumptions (6)

domain assumption Composition principle: serial composition of possible tasks is possible.
Used to argue the null task must be possible; this is a conjectured principle of constructor theory, stated as a physical principle in Section 3.
domain assumption Constructor-theoretic locality principle, including possibility of isolated substrates.
Invoked implicitly in Section 3 and used to infer possibility of isolated substrates in Section 4.
domain assumption Existence of a measure of accuracy and reliability relative to a subsidiary theory.
Section 2 assumes such a measure exists for any compliant subsidiary theory; the exact definition is left to the subsidiary theory.
ad hoc to paper Existence of timer-supporting theories, including a null constructor with attributes 0, R, 1 and a halt flag.
The core construction in Sections 5-6 presupposes an isolated substrate that spontaneously changes its own attributes; no proof from more basic principles is given.
domain assumption Poincare recurrence can be ignored, and finite systems approximate static attributes arbitrarily closely.
Footnote 3 in Section 6 relies on physical approximation to make static attributes well-defined.
domain assumption Real variables are parameterized by a continuous label lambda and the incremental ratio has a limit.
Section 7 constructs d v/d lambda by taking the limit Delta lambda -> 0; this presupposes differentiability, which must be supplied by subsidiary physics.

invented entities (1)

Null constructor (timer substrate with attributes 0, R, 1 and halt flag)
purpose: Provides a constructor-theoretic device whose completion signal defines duration equivalence classes.
No empirical prediction outside the paper is given; it is an idealization introduced to carry the notion of duration. The paper does not show such a substrate exists in any known physical theory beyond plausibility.

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Cite this review

Pith. "Pith review of Constructor theory of time." pith.science (2026). https://pith.science/paper/BOAVLGV3

@misc{pith2026250508692,
  author       = {Pith},
  title        = {Pith review of: Constructor theory of time},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BOAVLGV3}},
  note         = {Machine review of arXiv:2505.08692}
}

read the original abstract

Constructor theory asserts that the laws of physics are expressible as specifications of which transformations of physical systems can or cannot be brought about with unbounded accuracy by devices capable of operating in a cycle ('constructors'). Hence, in particular, such specifications cannot refer to time. Thus, laws expressed in constructor-theoretic form automatically avoid the anomalous properties of time in traditional formulations of fundamental theories. But that raises the problem of how they can nevertheless give meaning to duration and dynamics, and thereby be compatible with traditionally formulated laws. Here we show how.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Constructor-Theoretic Optical Time: Delay, Phase, and Fisher Distinguishability as Physical Tasks
physics.optics 2026-06 unverdicted novelty 6.0 of 10

The authors reformulate optical time using constructor theory, interpreting Fisher information as a distinguishability resource and the Cramer-Rao bound as a statement of task impossibility for delay estimation under ...
Tests of constructor theory
quant-ph 2026-06 unverdicted novelty 2.0 of 10

A review summarizing experimental proposals for testing constructor theory principles and their implications for physics.

Reference graph

Works this paper leans on

9 extracted references · 8 canonical work pages · cited by 2 Pith papers

[1]

This dependence is typically expressed in the form of differential equations in which time is an independent variable

Introduction: why time is problematic In the traditional conception of laws of physics, time appears as a real-valued parameter on which physical quantities depend. This dependence is typically expressed in the form of differential equations in which time is an independent variable. Yet this variable is not itself a physical quantity in the usual sense. F...

work page 2012
[2]

So, in particular, probability, which requires multiple outcomes to be possible, has no exact physical meaning under constructor theory

Inaccuracy, unreliability and limits Constructor theory requires perfect determinism in the sense that whatever transformation a given device would effect on a given object in a given state, it would also effect it on any other object of the same constitution in the same state. So, in particular, probability, which requires multiple outcomes to be possibl...

work page 1975
[3]

with attributes 𝑥 and 𝑦 to be jointly an input of a task, we can regard them as a single, composite substrate 𝒫!⊕𝒫

Possible and impossible tasks An attribute of a physical system is a set of some of the system’s possible states. Laws of physics under constructor theory are expressed in terms of tasks. In this paper it will suffice to consider a task to be an ordered pair of attributes 𝑥 and 𝑦 of a physical system 𝒫, which we write as 𝑥→𝑦 on 𝒫. (1) 𝒫 with attribute 𝑥 i...

work page 2015
[4]

In the traditional conception of physics, an isolated system is one whose dynamical interactions do not depend on variables of other systems

Isolated substrates and static attributes The foundation of the theory of time in constructor theory is the notion of an isolated substrate. In the traditional conception of physics, an isolated system is one whose dynamical interactions do not depend on variables of other systems. In constructor theory, we state this property without referring to time or...

work page
[5]

This task has no constraints on what substrate it is to be performed, nor does it have constraints on its allowed input or required outputs

The null task and constructors for it Consider the null task { }, which is represented by the empty set. This task has no constraints on what substrate it is to be performed, nor does it have constraints on its allowed input or required outputs. It is therefore the most lenient task. Remarkably, the composition principle implies that the null task must be...

work page 2015
[6]

Timers In the traditional conception, a timer with duration 𝑡 is a device that can be used to determine whether a physical process lasts more or less as long as the fixed interval 𝑡 characteristic of the timer. An idealised example in the traditional conception is an infinite straight line labelled by integers a distance 𝑑 apart, along which a classical p...

work page
[7]

Dynamics In order to recover dynamics as in the traditional conception, one needs to identify the law specifying how given variables of an isolated substrate 𝒫 change relative to the labels of a variable of a class of timers. Specifically, to express differential equations for physical variables in constructor-theoretic terms, we define (say) a real varia...

work page
[8]

Relation to ‘timeless’ theories of time in the traditional conception The peculiarities of time that we mentioned in Section 1 have inspired some physicists to seek ‘timeless’ formulations of the laws of physics – ones in which no time parameter appears explicitly and only the states of clocks do. There are such formulations for quantum theory (Page & Woo...

work page 1983

Show all 9 references

[9]

The trinity of relational quantum dynamics

Conclusions We have proposed a theory of clocks and timers in constructor theory, providing conditions for timer-supporting theories that are constructor-theory compliant. Then we have shown how in such theories one could recover dynamics as emerging from the timeless principl...

2007 arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

This dependence is typically expressed in the form of differential equations in which time is an independent variable

Introduction: why time is problematic In the traditional conception of laws of physics, time appears as a real-valued parameter on which physical quantities depend. This dependence is typically expressed in the form of differential equations in which time is an independent variable. Yet this variable is not itself a physical quantity in the usual sense. F...

work page 2012

[2] [2]

So, in particular, probability, which requires multiple outcomes to be possible, has no exact physical meaning under constructor theory

Inaccuracy, unreliability and limits Constructor theory requires perfect determinism in the sense that whatever transformation a given device would effect on a given object in a given state, it would also effect it on any other object of the same constitution in the same state. So, in particular, probability, which requires multiple outcomes to be possibl...

work page 1975

[3] [3]

with attributes 𝑥 and 𝑦 to be jointly an input of a task, we can regard them as a single, composite substrate 𝒫!⊕𝒫

Possible and impossible tasks An attribute of a physical system is a set of some of the system’s possible states. Laws of physics under constructor theory are expressed in terms of tasks. In this paper it will suffice to consider a task to be an ordered pair of attributes 𝑥 and 𝑦 of a physical system 𝒫, which we write as 𝑥→𝑦 on 𝒫. (1) 𝒫 with attribute 𝑥 i...

work page 2015

[4] [4]

In the traditional conception of physics, an isolated system is one whose dynamical interactions do not depend on variables of other systems

Isolated substrates and static attributes The foundation of the theory of time in constructor theory is the notion of an isolated substrate. In the traditional conception of physics, an isolated system is one whose dynamical interactions do not depend on variables of other systems. In constructor theory, we state this property without referring to time or...

work page

[5] [5]

This task has no constraints on what substrate it is to be performed, nor does it have constraints on its allowed input or required outputs

The null task and constructors for it Consider the null task { }, which is represented by the empty set. This task has no constraints on what substrate it is to be performed, nor does it have constraints on its allowed input or required outputs. It is therefore the most lenient task. Remarkably, the composition principle implies that the null task must be...

work page 2015

[6] [6]

Timers In the traditional conception, a timer with duration 𝑡 is a device that can be used to determine whether a physical process lasts more or less as long as the fixed interval 𝑡 characteristic of the timer. An idealised example in the traditional conception is an infinite straight line labelled by integers a distance 𝑑 apart, along which a classical p...

work page

[7] [7]

Dynamics In order to recover dynamics as in the traditional conception, one needs to identify the law specifying how given variables of an isolated substrate 𝒫 change relative to the labels of a variable of a class of timers. Specifically, to express differential equations for physical variables in constructor-theoretic terms, we define (say) a real varia...

work page

[8] [8]

Relation to ‘timeless’ theories of time in the traditional conception The peculiarities of time that we mentioned in Section 1 have inspired some physicists to seek ‘timeless’ formulations of the laws of physics – ones in which no time parameter appears explicitly and only the states of clocks do. There are such formulations for quantum theory (Page & Woo...

work page 1983