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Math-2848 · mathematics · Pde

A fixed-particle halting detector with time-square-integrable fluid derivatives

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every fixed positive computable viscosity, can each machine and finite input be effectively assigned smooth mean-zero forcing on $T^3$ with a unique global classical viscous flow so that every mixed derivative of force and velocity has time-dependent supremum norm in both $L_infinity(0,infinity)$ and L2(0,infinity), while a particle from a fixed initial label enters a fixed open strip exactly on halting? A divergence-free force may be required with the same velocity.

Short English statement

For every fixed positive computable viscosity, can each machine and finite input be effectively assigned smooth mean-zero forcing on $T^3$ with a unique global classical viscous flow so that every mixed derivative of force and velocity has time-dependent supremum norm in both $L_infinity(0,infinity)$ and L2(0,infinity), while a particle from a fixed initial label enters a fixed open strip exactly on halting? A divergence-free force may be required with the same velocity.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2848-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2848

Fixed-particle halting with bounded-energy derivatives: Every time and space derivative is required to have a bounded and square-integrable-in-time spatial supremum. The fixed initial particle label and detector are preserved for every machine input, including the option of solenoidal forcing with an unchanged velocity.

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

A fixed-particle halting detector with time-square-integrable fluid derivatives

OpenAI's claim

Fixed-particle halting with bounded-energy derivatives: The paper claims a smooth forced Navier--Stokes realization in which a fixed particle reaches a fixed open detector exactly when a prescribed machine halts. The fluid begins at rest on a flat three-torus, and mixed derivatives of velocity and force are bounded and square-integrable in time.

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. From any deterministic one-tape Turing machine $M$ and finite input $w$, the claim effectively constructs a smooth mean-zero force on the unit torus with a unique global smooth solution. Given any spatial derivative multi-index $\alpha$ and temporal derivative order $h$, the two functions $t\mapsto\|\partial_t^hD_x^\alpha u(t)\|_\infty$ and $t\mapsto\|\partial_t^hD_x^\alpha f(t)\|_\infty$ lie in $L^\infty(0,\infty)\cap L^2(0,\infty)$. With $\dot X=u(t,X)$ and $X(0)=(1/8,3/8,0)$, \[M(w)\text{ halts}\quad\Longleftrightarrow\quad X(t)\in\{1/2<x_1<1\}\text{ for some finite }t.\] A divergence-free force can be chosen with the same velocity and observation; after initial loading the velocity mechanism depends on $M$ alone.

Outline of the argument

Fixed-particle halting with bounded-energy derivatives: Finite branch labels are recorded as successive digits of a third-coordinate history. Smooth periodic gates read the current digit while earlier digits become invisible to that gate but remain stored. Longer physical steps compensate for increasingly fine spatial gates, and an all-time observation argument makes particle entry equivalent to halting throughout intermediate motions.

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.

Fixed-particle halting with bounded-energy derivatives: 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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