A closure criterion for when physical systems actually compute

What the work claims

This is a theoretical and formal paper, a position and framework piece rather than a new experiment, and it should be read as an argument to be evaluated on its logic rather than as a reported measurement. Its motivating case is a real experimental system: a droplet bouncing on a vertically vibrated fluid bath excites standing waves at each impact, and because those waves decay slowly, the wave field carries a finite-time memory of the droplet's past positions. In a published experiment by Perrard, Fort, and Couder, imposing a deliberate phase shift on that wave field causes the droplet to transiently retrace its own past trajectory while the newly written waves destructively interfere with, and erase, the old memory trace. The author uses this system, and the temptation to call it a "wave-based Turing machine," as the test case for a more general question: when does a physical system that writes, stores, reads, and erases information actually compute, as opposed to merely behaving in a way an observer can narrate computationally?¹

The claim is bold precisely because it cuts against a lot of loose language in physical and biological computing, including organoid computing. The paper's answer is a "closure criterion": physical memory and feedback are necessary but not sufficient for autonomous computation. What is additionally required is that an internal physical readout state in the system itself selects the system's next operation, rather than that operation being supplied from outside by an experimenter, a clock, or an external decoder. Applying this criterion to its own motivating example, the paper concludes that the wave-particle walker is a genuine "wave-memory physical machine with Turing-like primitives," but not a closed autonomous physical computer, because the phase shift that triggers erasure is imposed externally rather than selected by the system's own readout state.¹

How it works

The paper separates three levels that it argues are routinely conflated. The first is physical memory: a physical state that encodes information about the system's past, which the wave field clearly does. The second is transition-preserving computation: there exists a coarse-graining map from physical states to a smaller set of abstract states such that physical evolution "descends" cleanly to an abstract transition rule, meaning the coarse abstract description stays consistent as the physical system evolves. This requires more than an observer's convenient labeling; it requires that abstract states correspond to separated, noise-robust physical basins, and that the physical dynamics reliably carries each basin into the correct successor basin. The third, and strictest, level is closure: the system must contain an internal physical readout-control state whose value selects which physical operation happens next, expressed formally as a rule where the next state depends on an operation chosen by an internal readout of the current state, rather than an operation supplied from outside the system.¹

Applied to the walker, this becomes concrete. The wave field is a reservoir: a high-dimensional physical medium that stores a history and is sampled locally by the droplet through the wave's slope at the droplet's position, and that readout does influence the droplet's subsequent path, satisfying feedback. But the operation that produces the striking erasure result, the imposed phase shift, is not selected by any physical state internal to the walker; it is a parameter the experimenter sets from outside. The paper's proposed constructive fix is explicit and testable: couple a physical detector of wave-memory energy, local wave slope, or the droplet's trajectory state directly to the forcing phase, so that when the internal readout crosses a threshold, the system triggers its own phase shift and erases its own memory, with no outside intervention. Only that closed version would satisfy the criterion. The same logic generalizes to reservoir computing broadly: in standard physical or neuromorphic reservoir computing, a high-dimensional dynamical substrate transforms input histories into a rich internal state, but the readout that extracts useful output is typically an external, separately trained decoder, mathematically or electronically outside the reservoir itself, which is exactly the pattern the closure criterion flags as memory-with-feedback rather than closed autonomous computation.¹

The paper's own discussion section applies this directly to biological neural tissue, specifically cortical traveling waves, which is the most load-bearing part of the argument for organoid intelligence. It states plainly that "the presence of a traveling wave is not, by itself, a demonstration of computation," and proposes that a wave becomes computationally relevant only when one can identify the physical variables it transforms, the coarse-grained states it helps stabilize or route, and the downstream readout and control pathways through which it changes later neural dynamics. The author is careful to bound the analogy: the claim is not that cortex is a hydrodynamic walker, only that both systems share the structural pattern of distributed wave-like activity, local readout, and recurrent physical coupling, and the same closure question, whether the readout is internal and feeds back into control, applies to both.¹

Where a skeptic should push

The single most load-bearing move in the paper is treating "an internal physical readout state selects the next operation" as both necessary and, in combination with robust coarse-graining, sufficient for calling something an autonomous computer. That is a strong, restrictive definition, and the paper is honest about the cost: it explicitly declines to claim this is a complete theory of computational power, efficiency, or universality, and it acknowledges that in high-dimensional biological or fluid systems the relevant readout-control variables may be difficult to identify or measure at all, which makes the criterion easy to state and hard to operationalize experimentally for exactly the class of systems, organoid tissue included, where it would be most useful.

The paper is also, by design, a theoretical framework with one small worked physical example (the walker) and no new experimental test of the criterion itself; the "constructive extensions," an autonomous wave-memory eraser, a multistable readout trap, a coupled-walker controller, are explicitly proposed as future work, not built or tested here. A reader should therefore treat the criterion as a rigorous definition and a research program, not as a result that has itself been validated against data. There is also a real risk of the criterion proving too strict in one direction while still being too permissive in another: it is not obvious, and the paper does not attempt to show, that any current engineered neuromorphic reservoir, biological or silicon, actually satisfies full closure, which raises the question of whether the bar is set so high that essentially nothing built today clears it, making the criterion more useful as a diagnostic for what is missing than as a pass or fail test that sorts existing systems into two clean bins.

Does organoid computation pass its own closure test?

The non-obvious implication for organoid intelligence is that this criterion, applied honestly, is uncomfortable for the field's own flagship demonstrations. Closed-loop organoid computing systems, in their general published architecture, encode a task's sensory or reward signal as an electrical stimulation pattern delivered to the culture, record the organoid's spiking response on a microelectrode array, decode that response into a "prediction" or "action" using an external, computer-implemented decision rule, and then update the next stimulation pattern based on that external rule, closing the loop entirely outside the tissue. That architecture is structurally identical to the walker before the paper's proposed fix: a rich physical reservoir (the organoid), a local physical readout (the MEA), and an operation-selection step, choosing the next stimulus, that lives in the experimenter's code rather than in any physical state internal to the organoid. By the closure criterion developed here, that is memory and feedback with an externally imposed operation, not yet a closed autonomous physical computer, no matter how sophisticated the decoded behavior looks from the outside.

The genuine opportunity is that the paper turns a vague hype-correction into a concrete engineering target. It states explicitly that the closure criterion "suggests a stronger benchmark" for neuromorphic and physical reservoirs: whether the readout can be physically internalized so that decoded states control subsequent reservoir dynamics, rather than merely being measured. For organoid platforms, that reframes the research question productively: instead of asking whether an organoid can be trained to solve a task through an externally closed loop, which many published systems already do, it asks whether the organoid's own local physical or chemical readout structures, activity-dependent synaptic plasticity, local neuromodulatory release, or engineered optogenetic or chemogenetic feedback elements, can be made to select the tissue's next stimulation or plasticity event without leaving the biological substrate. That is a genuinely different and more ambitious design target than task accuracy under an external control loop, and it points toward a concrete near-term experiment: build the organoid analog of the paper's proposed autonomous eraser, a closed-loop element where a biologically local readout (for instance, a threshold-crossing local field potential or a genetically encoded activity sensor) directly triggers the tissue's own next perturbation, and test whether removing the external decision step changes what the system can do.

The genuine threat, and the sharper hype-correction, is that this criterion gives critics and regulators a rigorous, citable standard against which today's "organoid computes" and "biological intelligence" claims can be tested and often found wanting. The paper's own worked case, that a system with real memory, real feedback, and a visually striking behavioral signature (retracing a trajectory) still fails closure because the trigger is externally imposed, maps uncomfortably well onto organoid demonstrations where a compelling decoded behavior (learning a game, adapting to a stimulus statistics) is produced by a loop whose decision-making step lives entirely in silicon outside the dish. If the field continues to describe such systems as "computing" or "learning" without being precise about where operation-selection actually happens, this paper supplies exactly the vocabulary, the coarse-graining map, the robust basin, the internal readout-control variable, needed to show, dataset by dataset, that the tissue itself has not yet closed that loop. Used honestly, that is a discipline the field needs; used dishonestly by either side, it could become a rhetorical weapon that either overclaims (treating any local plasticity as sufficient for closure) or underclaims (demanding a bar so strict that it also disqualifies biological cortex, which the paper's own cortical-waves discussion suggests is a live and unresolved question, not a settled one).

The bottom line

Established, as a matter of definition and argument rather than measurement: the paper gives a coherent, restrictive, and useful formal distinction between physical memory-with-feedback and closed autonomous physical computation, and shows, using a real published hydrodynamic experiment, that a system with genuine writing, storage, reading, and erasure can still fail the stricter bar because the operation that triggers a key transition is externally imposed. Hypothesis, not yet demonstrated: that any specific organoid computing platform, or any specific instance of cortical wave activity, does or does not satisfy closure; the paper explicitly stops short of adjudicating biological cases and proposes cortical traveling waves as an open "test case," not a settled example either way. It is also not yet demonstrated that a physically closed version of the walker, or an analogous closed-loop organoid system, is buildable, only that the paper specifies what such a system would need to contain. What would confirm the framework's practical value is a built system, walker-based or biological, whose internal readout state is shown experimentally to select its own next operation, with the closed and externally-triggered versions compared directly. What would undercut it is a demonstration that the closure criterion, applied rigorously, either excludes systems (such as intact biological cortex) that any reasonable definition should call computational, or fails to exclude systems, such as a thermostat, that no one wants to call an autonomous computer, either of which would show the formal criterion has not yet found the right level of stringency.