we-solve (beta)

Math-2852 · mathematics · Pde

Halting detection by a fixed particle under initialized periodic solenoidal forcing

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For each fixed positive computable viscosity, can every deterministic one-tape machine and finite input be effectively encoded into smooth divergence-free mean-zero forcing on $T^3$, with a unique zero-data global classical Navier–Stokes solution, all mixed force and velocity derivatives globally bounded and uniform kinetic energy, such that both fields are one-periodic after initialization and the particle from one fixed label enters one fixed open strip exactly if the machine halts? Require the repeated mechanism for a fixed machine to be independent of its input.

Short English statement

For each fixed positive computable viscosity, can every deterministic one-tape machine and finite input be effectively encoded into smooth divergence-free mean-zero forcing on $T^3$, with a unique zero-data global classical Navier–Stokes solution, all mixed force and velocity derivatives globally bounded and uniform kinetic energy, such that both fields are one-periodic after initialization and the particle from one fixed label enters one fixed open strip exactly if the machine halts? Require the repeated mechanism for a fixed machine to be independent of its input.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2852-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2852

Reciprocal instructions by shear flows: The force must be divergence free and have mean zero, with the solution starting at rest. The initialized periodic mechanism has uniform mixed derivative and kinetic-energy bounds and realizes the fixed particle observation continuously, rather than only at instruction endpoints.

Geometric sheet programs and solenoidal forcing: The force must be divergence free and have mean zero, with the solution starting at rest. The initialized periodic mechanism has uniform mixed derivative and kinetic-energy bounds and realizes the fixed particle observation continuously, rather than only at instruction endpoints.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Halting detection by a fixed particle under initialized periodic solenoidal forcing

OpenAI's claim

Reciprocal instructions by shear flows: The paper claims smooth incompressible realization of finite reciprocal affine instructions on a flat three-torus; its recorder application detects halting by one particle with zero initial velocity and pressure and a solenoidal mean-zero force periodic after loading.

Geometric sheet programs and solenoidal forcing: The manuscript claims positive diagonal instruction maps on coding sheets realized by smooth incompressible flows, including exact compensation for planar area changes; it also gives a particle halting detector with mean-zero forcing periodic from time zero.

Mathematical statement of the claimed result

Mathematical claim 1.
For the incompressible system $\partial_tu+(u\cdot\nabla)u=-\nabla p+\nu\Delta u+f$, $\nabla\cdot u=0$, $u(0)=0$, choose viscosity $\nu>0$, whose value is computable. On the unit torus, an effective force from a one-tape machine with head moves $-1,0,1$ and finite input is divergence free and mean zero, and gives a unique global smooth solution $(U,0)$. All mixed derivatives are globally bounded, energy is uniformly bounded, and $U,f$ are one-periodic for $t\ge1$ and vanish near every integer time. Once loading is over, the repeating mechanism is specified by the machine alone; the input no longer alters that mechanism. The fixed particle $a_*=(1/8,3/8,1/2)$ enters $\{1/2<x_1<1\}$ exactly when the machine halts, including initial halting. The formulas remain valid relative to any fixed noncomputable real $\nu>0$.
Mathematical claim 2.
For the incompressible system $\partial_tu+(u\cdot\nabla)u=-\nabla p+\nu\Delta u+f$, $\nabla\cdot u=0$, $u(0)=0$, choose viscosity $\nu>0$, whose value is computable. The torus force is effectively prescribed from the finite machine table and input, divergence free, mean zero and one-periodic from time zero. The unique global smooth velocity has zero pressure, period one, vanishes near integer times, and obeys $(u\cdot\nabla)u=0$, uniformly bounded energy and globally bounded mixed derivatives. With $a_*=(1/4,1/2,1/4)$ and $O=\{1/2<x_1<7/8\}$, \[\exists t<\infty:X(t;a_*)\in O\quad\Longleftrightarrow\quad M(w)\text{ halts}.\] Initial halting is included; the construction can activate one Cartesian velocity component at a time.

Outline of the argument

Reciprocal instructions by shear flows: Separated heights resolve possible source--target overlap for the instruction rectangles. Reciprocal planar scaling is realized by localized shear motions on whole closed rectangles. Endpoint-size bounds control intermediate excursions independently of the scaling factor, while the reversible recorder and initialization place the particle in the correct code and preserve the detector corridor.

Geometric sheet programs and solenoidal forcing: Sheet maps are decomposed into localized shears and a normal correction that preserves volume. Explicit local inverses and private geometric positions retain erased planar information. Initialization and intermediate-trajectory estimates connect the finite rule table to the particle detector, and the prescribed divergence-free velocity gives the Navier--Stokes force by its exact residual.

Formal verification

Family 376 describes Lean coverage for the selected sheet-program, rapidly decaying particle-detector, velocity-detector, and full-box transport statements described separately in its Comparator files.

Reciprocal instructions by shear flows: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

Geometric sheet programs and solenoidal forcing: The manuscript links BalancedBoxRouting.lean, BalancedThreeStack.lean, ForcedNavierStokesComputation.lean, NavierStokesAlternating.lean, NavierStokesVelocity.lean, SolenoidalSheetPrograms.lean.

The documented selection does not cover the entire manuscript assertion relevant here.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

Go to contribution →