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Math-2839 · mathematics · Pde

Halting detection with one additional decay power per time derivative

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
At fixed positive computable viscosity, can every machine and input be effectively encoded into smooth mean-zero forcing and a unique zero-data global smooth Navier–Stokes flow on $T^3$, both obeying $||partial_t^j$ $D_x^\alpha $ $field(t)||_infinity\leq C_{j,\alpha }(1+t)^{-1-j}$, while one fixed particle enters one fixed strip exactly on halting? Permit either a zero-pressure force or its divergence-free projection with unchanged velocity.

Short English statement

At fixed positive computable viscosity, can every machine and input be effectively encoded into smooth mean-zero forcing and a unique zero-data global smooth Navier–Stokes flow on $T^3$, both obeying $||partial_t^j$ $D_x^\alpha $ $field(t)||_infinity\leq C_{j,\alpha }(1+t)^{-1-j}$, while one fixed particle enters one fixed strip exactly on halting? Permit either a zero-pressure force or its divergence-free projection with unchanged velocity.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2839-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2839

Scalar potentials and slow-clock computation: The slow-clock torus realization retains one fixed particle and strip while adding a decay power for each time derivative. The optional divergence-free projection alters the pressure representation while keeping the same velocity and observation event.

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

Halting detection with one additional decay power per time derivative

OpenAI's claim

Scalar potentials and slow-clock computation: The paper claims compactly supported forced fluid computation with a fixed particle and half-space detector, with time derivatives decaying at the stated inverse-time rates. Other realizations include invariant-plane instructions and velocity-field detection under a companion diffusion statement.

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. The force is smooth and effectively constructed on $\mathbb T^3$, with spatial mean zero. Its corresponding velocity is unique and globally smooth, and both fields satisfy \[\|\partial_t^jD_x^\alpha f(t)\|_\infty+\|\partial_t^jD_x^\alpha u(t)\|_\infty\le C_{j,\alpha}(1+t)^{-1-j}\quad(j\ge0).\] For initial particle position $(1/4,1/2,1/2)$, the event of entering $\{1/2<x_1<1\}$ is equivalent to halting of the supplied machine. A first force choice has zero pressure; its solenoidal projection gives a divergence-free choice with unchanged velocity and altered normalized pressure. Only loading depends on the input for a fixed machine.

Outline of the argument

Scalar potentials and slow-clock computation: Three scalar potentials successively lift a coded rectangle, apply its rule, and lower the image. A logarithmic time change remains onto an unbounded computational-time interval while reducing all time derivatives. Area-changing prefix maps are balanced by a normal component, and the alternative planar processor provides the stated stationary, periodic, and diffusion-based detector variants.

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.

Scalar potentials and slow-clock 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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