Theory of Computation

Motivation

Volume-preserving box transport supplies geometric operations for a finite computation. The selected goal assembles these into a force and a fixed particle observer. The pinned manuscript supplies the research context.

Setting

A finite machine and input determine effective velocity and force coefficients. The force depends affinely on positive viscosity, and after time one the velocity is periodic and depends only on the machine.

Formalization target

The selected goal is OAI.BalancedTransport.balanced_three_stack_realization. Its central assertion is

$$\exists t\geq0:\ X(t,(4,0,0))\in(-1,2)^3\quad\Longleftrightarrow\quad M\text{ halts on }w.$$

The theorem states that there exist velocity fields U(M,w), force coefficients f₀(M,w) and f₁(M,w), material flows X(M,w), and a globally 1-periodic velocity field R(M) for each finite deterministic tape machine M, with the following properties for every finite input word w. On nonnegative time and ℝ³, U, f₀ and f₁ are smooth and vanish outside a common spatial compact set independent of time; every mixed space-time derivative of U is uniformly bounded. Moreover, f₀ = ∂ₜU + (U·∇)U and f₁ = −ΔU. All three families are uniformly effective from the encoded machine and input: algorithms approximate every mixed derivative component at rational space-time points within 2⁻ⁿ, provide global integer bounds on these derivatives, and provide integer support radii. The velocity is 1-periodic after time 1 and satisfies U(M,w)(t,x) = R(M)(t,x) for every t ≥ 1, so its later field depends only on M. The flow satisfies X(0,a) = a and ∂ₜX(t,a) = U(t,X(t,a)); its trajectory starting at (4,0,0) enters (−1,2)³ at some nonnegative time exactly when M halts on w, meaning that a configuration with no next transition is reached after finitely many machine steps. For every viscosity ν > 0, the force f = f₀ + νf₁ is smooth, has uniformly bounded mixed derivatives, and is 1-periodic after time 1; U with pressure zero solves the incompressible Navier–Stokes equation ∂ₜU + (U·∇)U = νΔU + f with U(0,·) = 0. This velocity is unique among smooth zero-data solutions in the comparison class, and every such solution has spatially constant pressure. The comparison class requires the velocity and all spatial derivatives through order two to be continuous in time as L² functions, the velocity to be continuously differentiable in time in L², velocity and first spatial derivatives to be bounded on each finite time slab, and pressure, after subtracting a time-dependent scalar, together with its first spatial derivatives to be continuous in time in L²; (U,0) belongs to this class. Finally, derivative approximations and global derivative bounds for f are computable when ν is computable, and computable relative to any rational name of ν whose nth approximation has error at most 2⁻ⁿ.

Significance and status

The balanced three-stack realization is the main goal; balanced box routing is a separate supporting reference. The exact conclusion permits spatially constant pressure in competing solutions and explicitly states its comparison class. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.

Difficulty

Uniform effectiveness, compact support, periodic behavior and uniqueness must all coexist with exact halting detection.

Formalization scope

The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.

Additional published targets are included as separate references:

• OAI.BoxTransport.Routing.balanced_box_routing (Open).

Selected references

• OpenAI, Incompressible Box Transport and Finite Computation, preprint, 2026. Manuscript.
• OpenAI, accompanying formal statements, commit adc7f1241b42. Selected goal source.

Motivation

A force whose derivatives decay rapidly can still encode a computation in a particle trajectory. The selected construction makes the observer condition and effectiveness requirements precise. The pinned manuscript supplies the research context.

Setting

For each finite deterministic machine and valid input, compilers produce smooth force and velocity fields on nonnegative time and real three-space, with common compact spatial support.

Formalization target

The selected goal is OAI.AlternatingNS.alternating. Its central assertion is

$$\exists t\geq0:\ X_1(t)>0\quad\Longleftrightarrow\quad M\text{ halts on }w.$$

The theorem states that for every positive real viscosity ν computable by rational approximations with error at most 2⁻ⁿ, there exist computable compilers for a force f and velocity U, and one compact set K⊂ℝ³, with the following property for every well-formed finite deterministic Turing machine M and valid finite input w. The compiled programs describe smooth fields f,U on [0,∞)×ℝ³, both spatially supported in K for all nonnegative times, such that every mixed space-time derivative D satisfies sup_{t≥0,x}(1+t+‖x‖)ᴶ‖Df(t,x)‖<∞ and the analogous bound for U, for every nonnegative integer J. Each program supplies rational approximations to all mixed derivatives at rational space-time points with requested error 2⁻ᵏ, effective moduli of continuity on bounded regions, and integer bounds for all these weighted derivative norms. The velocity starts from zero, is divergence-free, and solves the classical forced Navier–Stokes equation ∂ₜU+(U·∇)U=νΔU+f with identically zero pressure; specifically f=∂ₜU+(U·∇)U−νΔU. On every interval [0,T], U is continuous in H² and continuously differentiable in L², and U and its spatial derivative are uniformly bounded in space and time. Moreover, any classical solution (v,p) with the same force and zero initial velocity equals (U,0) pointwise for all nonnegative times, provided on every [0,T] it has v continuous in H², continuously differentiable in L², and uniformly bounded, and p continuous in H¹. Here these Sobolev continuity conditions mean continuous L² representatives of every spatial derivative through the indicated order; time derivatives at zero are taken within [0,∞). Every initial point a has a unique trajectory X(a,t) for t≥0 satisfying X(a,0)=a and ∂ₜX(a,t)=U(t,X(a,t)). Finally, the trajectory starting at (−1,0,0) has strictly positive first coordinate at some nonnegative time if and only if M halts on w, where encountering a missing instruction also counts as halting.

Significance and status

The goal is the alternating-coordinate construction at every positive computable viscosity. It includes Sobolev time regularity, effective bounds, zero initial velocity and the exact observer condition, rather than all three manuscript constructions. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.

Difficulty

The encoding must preserve all weighted derivative bounds and uniqueness in the stated comparison class while recording arbitrarily long computations.

Formalization scope

The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.

The available published material supplies this goal and its necessary definitions. Additional manuscript lemmas are not represented as attached milestones.

Selected references

• OpenAI, Computation under Rapidly Vanishing Navier–Stokes Forcing, preprint, 2026. Manuscript.
• OpenAI, accompanying formal statements, commit adc7f1241b42. Selected goal source.

Motivation

Incompressible transport can implement prescribed geometric operations on coding sheets. The selected statement concerns one such finite transport construction with effective smooth forcing. The pinned manuscript supplies the research context.

Setting

Source and target rectangles have rational data and specified positive diagonal ratios. In the formula, D_i(y)=q_i+diag(r_i)(y−p_i), with source and target centers p_i and q_i. Space is represented periodically with period ten, and the viscosity is a positive computable real.

Formalization target

The selected goal is OAI.Solenoidal.sheet_theorem. Its central assertion is

$$X(1,\mathrm{sheet}(y))=\mathrm{sheet}(D_i y)\pmod{(10\mathbb Z)^3}.$$

The theorem states that, for any natural number N, any sheet datum D of size N, and any computable real viscosity ν>0, there exist a forcing field f and a velocity field u (each a time-dependent vector field on ℝ³) and a flow map X such that the following hold. Here D consists of N source rectangles and N target rectangles with rational corners, all inside the square [2,3]², with the sources pairwise separated by a positive distance and likewise the targets, together with positive rational ratios in each of two coordinates such that the diagonal affine map from source i to target i, which sends the source center to the target center and scales offsets by the ratios, maps the source rectangle exactly onto the target rectangle. Both f and u are periodic in space with period 10 in every coordinate and are C^∞ in space and time jointly. The forcing f is effective, meaning that all its mixed space-time partial derivatives can be approximated to any rational accuracy by a single fixed partial recursive procedure from computable names of the evaluation point. Moreover f has zero mean over the fundamental cell [0,10)³ at every time, is divergence free, and is 1-periodic in time. The field u is a classical solution on t≥0 of the forced Navier–Stokes equations ∂ₜu + (u·∇)u = −∇p + ν Δu + f with zero pressure, zero initial data, incompressibility and spatial periodicity, so u is a solution driven by f without pressure. Any classical solution v with pressure p for the same f and ν and with zero initial velocity coincides with u, and has p identically zero, for all t≥0. The flow map X satisfies X(0,a)=a and solves the particle-path equation dX/dt=u(t,X) for t≥0, and for each i and each point y of the i-th source rectangle, the time-1 flow image of the sheet point (y₀,y₁,2) agrees modulo the torus ℝ³/(10ℤ)³ with the sheet point at the diagonal-map image of y. Further, u vanishes in a neighborhood of every integer time, the advection term (u·∇)u vanishes everywhere, and both f and u have all their mixed space-time derivatives uniformly bounded by rational bounds computable from the derivative multi-index by a fixed partial recursive procedure. Finally, if N=0 then f and u are identically zero.

Significance and status

The target is the sheet theorem, including its encoded uniqueness statement and zero-advection conditions. Machine-halting detection is not a conclusion of this published goal. The target is currently Open on Prove2Me. The manuscript's mathematical argument and a machine-checked proof of the selected statement are separate deliverables.

Difficulty

The construction must simultaneously enforce smoothness, incompressibility, temporal periodicity, effective bounds and exact transport on every point of each sheet rectangle.

Formalization scope

The exact published goal, its hypotheses and its referenced definition blocks specify the requested formalization. The explanatory formula above is a summary; all quantifiers and additional clauses in the linked statement remain required.

The available published material supplies this goal and its necessary definitions. Additional manuscript lemmas are not represented as attached milestones.

Selected references

• OpenAI, Geometric Programs for Solenoidal Forcing, preprint, 2026. Manuscript.
• OpenAI, accompanying formal statements, commit adc7f1241b42. Selected goal source.