Programming a viscous fluid to detect whether a computation halts

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Family 376 · Partial differential equations
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-26

1. The question

Start a viscous fluid at rest. Fix one marked particle and a region in space. Can a finite description of an external force make that particle enter the region exactly when a given computer program eventually stops?

This family of nine manuscripts claims several ways to do so in three-dimensional flat geometry. The force encodes the machine and its finite input. The fluid motion carries out the instructions, and entry into a fixed detector reports halting.

The resulting prediction problem is undecidable: no algorithm can take every force description in the constructed class and always answer correctly whether detection will occur. This is a statement about exact mathematical trajectories over arbitrarily long times. It does not mean the fluid computes answers to undecidable problems in finite time: a machine that never halts produces no finite-time halting signal.

The family also distinguishes several experiments. Some forces repeat periodically, another becomes stationary, and others decay. Most detectors follow a particle; one manuscript instead tests the evolving velocity. These alternatives have different guarantees, so they should not be combined into one imagined construction possessing every property at once.

2. Background and essential definitions

The fluid equation

For velocity $u(t,x)$, pressure $p(t,x)$, viscosity $\nu>0$, and external body force $f(t,x)$, the equations are

$$\partial_tu+(u\cdot\nabla)u=-\nabla p+\nu\Delta u+f, \qquad \nabla\cdot u=0.$$

The divergence condition expresses incompressibility. A solenoidal vector field is simply a divergence-free one. The nonlinear term is the acceleration due to particles moving through a spatially varying velocity; the Laplacian represents viscous diffusion.

Several constructions use the flat unit torus $\mathbb T^3=(\mathbb R/\mathbb Z)^3$: a cube with opposite faces identified. Others use $\mathbb R^3$ or $\mathbb R^2\times\mathbb T$. All begin with $u(0,x)=0$, and viscosity is fixed before the machine is supplied.

A particle with initial label $a$ follows

$$\dot X(t)=u(t,X(t)),\qquad X(0)=a.$$

The event “$X(t)\in O$ for some finite $t$” is a material-particle observation. An Eulerian observation instead inspects the velocity at locations in space, without following one label.

What counts as a finite program for a force?

A Turing machine has finitely many control states and instructions but an unbounded tape. A finite input uses only finitely many nonblank cells initially. Its halting problem asks whether its instructions eventually reach a designated stop state.

An effective smooth force is described by a finite program that evaluates the force and each requested derivative to arbitrary rational accuracy at computably specified arguments. The program also supplies the relevant derivative bounds. No efficiency bound is claimed.

Crucially, the construction installs all local instruction branches. It does not first discover the machine’s future execution and then replay a precomputed particle path. Unqualified effectiveness assumes a positive computable viscosity; for an arbitrary fixed real viscosity, evaluation is relative to access to that parameter.

Infinite information in bounded coordinates

A tape can be represented by a positional expansion. In base $B$, a word $a_0a_1\ldots$ can be encoded as

$$z=\sum_{j\geq0}d(a_j)B^{-j-1},$$

where the digits $d(a_j)$ represent symbols. Choosing separated digits leaves gaps between the intervals identifying different first symbols. Removing a first symbol becomes an affine expansion; inserting one becomes an affine contraction.

The unbounded tape is stored in arbitrarily fine digits of an exact real coordinate. This explains both the computational capacity and the absence of a uniform finite-precision guarantee.

3. Development history and previous partial results

Turing’s undecidability theorem supplies the logical obstruction. Moore’s work in 1990–1991 connected symbolic computation with generalized shifts, piecewise affine maps, and smooth dynamical systems. It established a route from local tape instructions to continuous trajectories.

There is an obstacle to using fluid trajectories: a smooth flow is invertible between finite times, while an ordinary instruction can erase information. Landauer discussed retaining intermediate information, and Bennett’s 1973 reversible simulation explicitly recorded instruction history. The present manuscripts use related retention principles, with several different placements of the record.

Fluid computation itself also has substantial predecessors. Cardona, Miranda, Peralta-Salas and Presas constructed Turing-complete stationary Euler flows on a three-sphere with an adapted Riemannian metric. Cardona, Miranda and Peralta-Salas obtained universal stationary Beltrami fields in Euclidean three-space; those universal fields have infinite energy and a noncompact computational set. Their work also gives robust tape-bounded simulations on the flat torus, including a viscous realization from nonzero initial velocity.

Dyhr, González-Prieto, Miranda and Peralta-Salas constructed stationary unforced Navier–Stokes computation with an adapted metric and Hodge viscosity. In that setting a harmonic velocity makes the viscous term vanish. These are important results, but the choices of geometry, initial data and observation differ from the fixed flat, externally forced, initially resting experiments here.

For velocity observations, earlier work by Cardona and collaborators encoded computation in the Euler evolution on a constructed higher-dimensional compact manifold. M7 changes the setting to positive viscosity in three flat dimensions and supplies pointwise or integral detectors.

The claimed advance is therefore a collection of explicit forced-fluid realizations under carefully separated constraints, rather than the first connection between computation and fluid dynamics.

4. What the new manuscripts claim

A common decision statement

Each particle construction fixes its geometry, viscosity, initial velocity, label $a$, and detector $O$ before receiving a machine $M$ and word $w$. It constructs a force with a unique global smooth solution in its declared comparison class such that

$$\bigl(\exists t<\infty:\ X(t,a)\in O\bigr) \quad\Longleftrightarrow\quad M\text{ halts on }w.$$

The fixed label and detector may differ between constructions. Uniqueness on the torus is among classical solutions on every finite time interval. Whole-space statements specify additional Sobolev and pressure conditions; they do not assert uniqueness without a declared solution class.

How the nine manuscripts fit together

Source	Main additional feature
M1, Finite Instructions and Solenoidal Shear Flows	Unit-torus particle detection with directly solenoidal, mean-zero forcing, zero pressure, and period one after initialization. All mixed derivatives are bounded.
M2, A Fixed Particle Test	The ordinary machine writes its history into a third coordinate. Longer steps compensate for finer gates, making every mixed derivative of force and velocity bounded and square-integrable in time in spatial supremum norm.
M3, Eventually Stationary Forcing	A spatial clock drives a finite mechanism. The mean-zero force becomes independent of time after time one; the fixed particle continues computing.
M4, Geometric Programs	Solenoidal mean-zero forcing is periodic from time zero, with zero initial velocity and pressure. Separate geometric results handle area-changing sheet maps and reciprocal maps on thick boxes.
M5, Prefix Instructions	Several history processors realize finite prefix replacements. Normal compensation handles changes of planar area; separate variants change support, initialization, clocks and detectors.
M6, Rapidly Vanishing Forcing	Three memory constructions give decay faster than every inverse power of time for every fixed derivative. Each machine step takes one unit of physical time, with progressively smaller spatial signals.
M7, Velocity-Field Detection	Halting is detected by a velocity component exceeding a fixed threshold on the torus, or by its integral on $\mathbb R^2\times\mathbb T$. The two regimes have different bounds and support properties.
M8, Scalar Potentials and Slow Clocks	A compactly supported whole-space particle construction has derivative decay $O((1+t)^{-1-j})$ for time-derivative order $j$. Other constructions provide invariant-plane and spatial-clock mechanisms.
M9, Incompressible Box Transport	Determinant-one diagonal maps are realized on neighborhoods of entire solid boxes. A balanced three-stack recorder gives compactly supported forcing, periodic after loading, with a fixed open cube as detector.

M9 is the October 6 manuscript; the other eight are dated September 27.

M7 deserves an additional qualification. On the torus it detects whether $u_3>1/2$ somewhere in a fixed strip at some time, but force derivatives need only be bounded on each finite time interval. On $\mathbb R^2\times\mathbb T$, all force derivatives are globally bounded, and the detector asks whether the integral of $u_3$ over a fixed half-space times the circle exceeds $1/2$. That construction expands spatially and claims neither fixed compact support for all time nor uniformly bounded total kinetic energy.

5. Why the advance matters

Importance: 4/5. If correct, this is a broad constructive contribution linking computability, incompressible geometry and dissipative PDEs. Its significance lies in explicit realizations with fixed flat geometry, zero initial velocity, and several sharply distinguished restrictions on the force and observer.

Smoothness and uniqueness do not guarantee that every long-term event is algorithmically decidable. A perfectly smooth, explicitly described solution can carry a decision problem whose answer is not uniformly computable over all descriptions.

The decaying constructions make this especially striking. A particle may travel only a finite total distance while carrying an unbounded computation in finer spatial scales. The exact observer is essential to that conclusion. The work does not establish a robust physical computer with unlimited storage at fixed measurement accuracy.

6. The main idea

Preserve information before moving fluid

Two tape halves occupy two positional coordinates. An instruction reads finitely many symbols, so it acts on an entire rectangle of codes sharing those prefixes. For reciprocal maps, expansion in one direction is balanced by contraction in the other.

Information that would be overwritten must remain somewhere. M1 and M3 use finite reversible recording tables. M2 stores branch indices in a third real coordinate and reads the newest digit through a fine periodic gate. M5 explores other prefix and history arrangements. M9 uses three stacks with balanced total prefix lengths, making the corresponding three-dimensional affine map have determinant one.

The distinction between injectivity on one chosen run and injectivity on the full allowed instruction domain matters: the latter separates target blocks for arbitrary unused tails.

Route whole neighborhoods and control the journey

Even when source blocks are mutually disjoint and target blocks are mutually disjoint, a target can overlap another source. Moving one block directly may disturb information waiting elsewhere.

The geometric constructions use private heights or temporary storage, followed by delivery along separated routes. Local curls and shear pulses preserve divergence-free motion. In M9, obstacle detours transport full solid boxes, not just a zero-thickness coding surface. When a planar map changes area, a compensating normal expansion or contraction preserves three-dimensional volume.

Endpoint correctness alone is insufficient: a nonhalting particle must avoid the detector throughout every intermediate motion. The manuscripts therefore include coordinate bounds, height guards or detours appropriate to each observer.

Convert the mechanism into a forced solution

Once a smooth divergence-free velocity $U$ has been prescribed, define

$$f=\partial_tU+(U\cdot\nabla)U-\nu\Delta U.$$

Then $(U,0)$ satisfies the fluid equations. The substantive construction is the effective computational velocity and its all-time detector bounds. A classical difference-energy estimate gives uniqueness for these constructed data in the specified classes.

For certain shear schedules, the active velocity is transverse to its own spatial variation, so $(U\cdot\nabla)U=0$. The resulting force is directly solenoidal with zero pressure. Other torus forces can be projected to their divergence-free part, changing the pressure while preserving velocity. This projection should not be confused with the direct shear property or assumed to preserve compact support in Euclidean space.

Slow time or shrink space

A clock $\tau(t)$ replaces a recurring velocity $V$ by

$$U(t,x)=\tau'(t)V(\tau(t),x).$$

It preserves the entire trajectory only if accumulated computational time remains unbounded. A logarithmic clock achieves decay of order $1/t$ while still traversing infinitely many steps.

Decay faster than every inverse power makes a nonnegative clock speed integrable, so this simple slowdown cannot drive infinitely many steps of the same recurring mechanism. M6 instead retains unit physical step lengths and shrinks the spatial displacement much faster than the cost of differentiating its selectors grows. Its three memories use moving localized curls, alternating fractional coordinates, or periodic lattice selectors. Late halting signals can be arbitrarily small.

Carry a signal through diffusion

M7 uses a velocity $U=(a,w)$ independent of the third coordinate, where $a$ is planar and divergence free. The vertical component satisfies

$$\partial_tw+a\cdot\nabla w=\nu\Delta w+h.$$

The horizontal velocity and scalar source $h$ are prescribed; the scalar $w$ is solved for. Thus the force does not conceal an unknown solution in its definition.

On the torus, increasingly short stirring bursts keep fresh signals close to pure transport, while waiting intervals dissipate older signals. In the expanding construction, a cutoff moving with the computational address controls escaped scalar mass. These two methods pay for diffusion with different resources.

7. Scope, limitations, and open questions

All conclusions concern specially constructed external forces. They give no general regularity theorem or blowup result for arbitrary three-dimensional Navier–Stokes data, and no resolution of the unforced global regularity problem.

The reductions use exact real encodings and unlimited observation time. They assert neither efficient simulation nor uniform tolerance to force perturbations, measurement error, or finite-precision replacement. Even M7’s strict detection thresholds do not themselves imply stability under arbitrary perturbations of the forcing.

Periodicity, eventual stationarity and decay are separate choices. Directly solenoidal forces, zero pressure, fixed compact support and globally bounded derivatives are asserted only where the individual construction supplies them.

A natural further direction is to understand which computational behaviors persist under specified noise or finite storage constraints. The existing robust tape-bounded predecessors illustrate why that is a different question from unrestricted exact universality.

8. Sources and further reading

• Bennett, Logical Reversibility of Computation: history retention in reversible simulation.
• Moore, Generalized shifts: unpredictability and undecidability in dynamical systems: positional coding and smooth dynamical realizations.
• Cardona, Miranda, Peralta-Salas and Presas, Constructing Turing complete Euler flows in dimension 3: adapted-geometry stationary Euler computation.
• Cardona, Miranda and Peralta-Salas, Computability and Beltrami fields in Euclidean space, version 3: Euclidean universality and the distinct tape-bounded torus results.
• Dyhr, González-Prieto, Miranda and Peralta-Salas, Turing complete Navier–Stokes steady states via cosymplectic geometry: the unforced construction with adapted geometry and Hodge viscosity.

Source manuscripts

• M1. OpenAI, Finite Instructions and Solenoidal Shear Flows (September 27, 2026).
• M2. OpenAI, A Fixed Particle Test for Computation in a Forced Viscous Flow (September 27, 2026).
• M3. OpenAI, Universal Computation with Eventually Stationary Navier–Stokes Forcing (September 27, 2026).
• M4. OpenAI, Geometric Programs for Solenoidal Forcing (September 27, 2026).
• M5. OpenAI, Prefix Instructions and Incompressible Flows (September 27, 2026).
• M6. OpenAI, Computation under Rapidly Vanishing Navier–Stokes Forcing (September 27, 2026).
• M7. OpenAI, Velocity-Field Detection of Computation in Forced Navier–Stokes Flows (September 27, 2026).
• M8. OpenAI, Scalar Potentials and Slow Clocks for Forced Fluid Computation (September 27, 2026).
• M9. OpenAI, Incompressible Box Transport and Finite Computation (October 6, 2026).

Sources consulted on 26 October 2026 at repository revision 1581d4c. Download the manuscript citations.