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Math-2840 · mathematics · Pde

Computation by a fixed particle with eventually stationary forcing

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
At every fixed positive computable viscosity, can a finite machine and input effectively determine smooth mean-zero forcing on $T^3$, independent of time after $t=1$ and with every derivative globally bounded, whose zero-data Navier–Stokes solution is globally smooth, classically unique and has uniformly bounded energy, while one fixed particle enters one fixed open strip exactly if the encoded machine halts?

Short English statement

At every fixed positive computable viscosity, can a finite machine and input effectively determine smooth mean-zero forcing on $T^3$, independent of time after $t=1$ and with every derivative globally bounded, whose zero-data Navier–Stokes solution is globally smooth, classically unique and has uniformly bounded energy, while one fixed particle enters one fixed open strip exactly if the encoded machine halts?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2840-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2840

Eventually stationary forced computation: The computational rule table supplies a finite effective force prescription, which is stationary after the loading interval. Bounded derivatives, classical uniqueness and uniformly bounded kinetic energy accompany the particle event; no trajectory-dependent choice of forcing is made.

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

Computation by a fixed particle with eventually stationary forcing

OpenAI's claim

Eventually stationary forced computation: The manuscript claims smooth mean-zero forcing on the flat three-torus that becomes stationary after time one and lets a fixed particle detect arbitrary machine halting. The force has a finite effective description and the fluid begins at rest.

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 flat unit torus, an effective smooth force from the finite machine and input is mean zero, independent of time for $t\ge1$, and has all mixed derivatives bounded globally. The zero-data solution is globally smooth and unique on finite classical time intervals, with uniformly bounded kinetic energy. The fixed particle $P=(1/4,1/4,0)$ enters $O=\{1/32<x_2<1/8\}$ at a finite time exactly when the machine halts. The finite force prescription does not execute the selected computation; the same formulas allow any positive viscosity parameter.

Outline of the argument

Eventually stationary forced computation: A reversible recording table is encoded by gapped tape rectangles. Ordered extraction into a parking region, reciprocal deformation, and delivery implement every branch without first running the computation. A third spatial coordinate drives the finite rule table autonomously; a fixed startup ramp changes rest into the autonomous field, after which the force is stationary.

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.

Eventually stationary forced computation: No manuscript-specific Comparator is listed in the pinned inventory.

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

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