LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Partition and quadrature bounds for a periodic PIC shape

Statement

Let $d, L, N, h, η_{ε}, Z_{ε, h}$ , and $w_{j}^{ε, h}$ have the meanings in Periodic uniform PIC grid and inner product and Periodic PIC deposition shape and gather operators. For a finite signed Borel measure $μ$ on the torus, write $∥ μ ∥_{T V} = ∣ μ ∣ (T_{L}^{d})$ and define $(P_{h}^{ε} μ)_{j} = h^{- d} \int_{T_{L}^{d}} w_{j}^{ε, h} (x) d μ (x) .$ Put $C_{η} = \frac{L^{d}}{12} \sum_{a = 1}^{d} ∥ \partial_{a a} η_{ε} ∥_{\infty}, Z_{\min} = L^{d} \min_{x \in T_{L}^{d}} η_{ε} (x) > 0.$ Then $\sum_{j} w_{j}^{ε, h} (x) = 1$ for every $x$ , and $Z_{ε, h} (x) \geq Z_{\min}$ . The measure assignment satisfies $∣ (P_{h}^{ε} μ)_{j} - (η_{ε} * μ) (x_{j}) ∣ \leq \frac{C_{η} ∥ η_{ε} ∥_{\infty}}{Z_{\min}} ∥ μ ∥_{T V} h^{2} .$ For $g \in C^{2} (T_{L}^{d}; R^{m})$ with finite $m \geq 1$ (including scalar $m = 1$ ), use the Euclidean norm $∣ \cdot ∣_{2}$ and $∥ g ∥_{\infty} = \sup_{y} ∣ g (y) ∣_{2}$ . Define the translation-uniform constant $C_{η, g} = \frac{L^{d}}{12} \sum_{a = 1}^{d} \sup_{x, y \in T_{L}^{d}} {∣ \partial_{y_{a} y_{a}} [η_{ε} (y - x) g (y)] ∣}_{2} .$ The gathered nodal samples satisfy the explicit bound $∣ \sum_{j} w_{j}^{ε, h} (x) g (x_{j}) - (η_{ε} * g) (x) ∣ \leq \frac{C_{η, g} + C_{η} ∥ g ∥_{\infty}}{Z_{\min}} h^{2} (x \in T_{L}^{d}) .$

Facts & Assumptions

Given: The periodic grid and pairing, the strictly positive normalized smooth kernel, its sampled normalizer, and the weights defined in Periodic uniform PIC grid and inner product and Periodic PIC deposition shape and gather operators.

[F1]

Periodic uniform PIC grid and inner product: $h = L / N$ , the grid is the uniform periodic tensor grid, and $h^{d} \sum_{j}$ is its volume-weighted sum.

[F2]

Periodic PIC deposition shape and gather operators: $Z_{ε, h} (x) = h^{d} \sum_{j} η_{ε} (x_{j} - x)$ and $w_{j}^{ε, h} (x) = h^{d} η_{ε} (x_{j} - x) / Z_{ε, h} (x)$ .

[F3]

Composite periodic trapezoidal error in a uniform tensor grid: for $f \in C^{2} (T_{L}^{d})$ , the uniform periodic tensor-grid trapezoid error is at most $(L^{d} h^{2} / 12) \sum_{a} ∥ \partial_{a a} f ∥_{\infty}$ .

[F4]

Periodic PIC regularization gives a smooth compatible Poisson solution: $η_{ε}$ has unit integral.

Proof

1.1F1F2algebra

Strict positivity of $η_{ε}$ implies $Z_{ε, h} (x) > 0$ . Summing the weights in [F2] gives $\sum_{j} w_{j}^{ε, h} (x) = Z_{ε, h} (x) / Z_{ε, h} (x) = 1$ , and each weight is nonnegative. Since every summand in $Z$ is at least $h^{d} \min η_{ε}$ , and $h^{d} ∣ J_{N} ∣ = L^{d}$ , $Z_{ε, h} (x) \geq Z_{\min}$ .

1.2F1F3F4algebra

For fixed $x$ , put $f_{x} (y) = η_{ε} (y - x) g (y)$ and let $q_{x} = h^{d} \sum_{j} f_{x} (x_{j}) - \int f_{x} (y) d y \in R^{m}$ . For every Euclidean unit vector $v$ , apply the scalar estimate [F3] to $v \cdot f_{x}$ . Since $∣ v \cdot \partial_{a a} f_{x} ∣ \leq ∣ \partial_{a a} f_{x} ∣_{2}$ , this gives $∣ v \cdot q_{x} ∣ \leq C_{η, g} h^{2}$ . If $q_{x} \neq 0$ , take $v = q_{x} / ∣ q_{x} ∣_{2}$ ; if $q_{x} = 0$ the bound is immediate. Thus $∣ q_{x} ∣_{2} \leq C_{η, g} h^{2}$ , with no dimension factor. The integral is $(η_{ε} * g) (x)$ because the kernel is even. Projecting that integral onto its unit direction in the same way, nonnegativity and the unit integral [F4] give $∣ (η_{ε} * g) (x) ∣_{2} \leq ∥ g ∥_{\infty}$ . The defining supremum is over both $x$ and $y$ : translation of the kernel alone does not translate $g$ .

2.1F1F2F3F4step 1.1algebra

Apply [F3] to $y \mapsto η_{ε} (y - x)$ ; translation does not change its second-derivative suprema and its integral is $1$ by [F4]. Thus $∣ Z_{ε, h} (x) - 1 ∣ \leq C_{η} h^{2}$ . By [F2], $(P_{h}^{ε} μ)_{j} = \int_{T_{L}^{d}} \frac{η_{ε} (x_{j} - x)}{Z_{ε, h} (x)} d μ (x) .$ Subtract $(η_{ε} * μ) (x_{j})$ , factor $η_{ε} (x_{j} - x)$ , and use $∣ Z^{- 1} - 1 ∣ = ∣ Z - 1 ∣ / Z \leq C_{η} h^{2} / Z_{\min}$ . Integrating against the total variation measure gives the first bound.

3.1F2step 1.1step 1.2algebra∎

The gathered sum in the Statement is $h^{d} \sum_{j} f_{x} (x_{j}) / Z_{ε, h} (x)$ by [F2]. Subtract the integral, use the quadrature error from step 1.2 divided by $Z \geq Z_{\min}$ , and bound the Euclidean norm of the normalization correction by $∣ Z^{- 1} - 1 ∣ ∥ g ∥_{\infty} \leq C_{η} h^{2} ∥ g ∥_{\infty} / Z_{\min}$ . This is the second bound.

Remarks

The shifted-product constant is finite and can be bounded entirely by separate derivative suprema: $C_{η, g} \leq \frac{L^{d}}{12} \sum_{a = 1}^{d} (∥ \partial_{a a} η_{ε} ∥_{\infty} ∥ g ∥_{\infty} + 2 ∥ \partial_{a} η_{ε} ∥_{\infty} ∥ \partial_{a} g ∥_{\infty} + ∥ η_{ε} ∥_{\infty} ∥ \partial_{a a} g ∥_{\infty}) .$ This follows by differentiating the product twice and using the Euclidean triangle inequality. The unshifted product $η_{ε} (y) g (y)$ alone does not supply a uniform bound for all gather locations $x$ .

Depends on

Used by

Fixed-regularization periodic PIC force consistency on a smooth density Theorem

Dependency tree · one level

4 results within one dependency step of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles, Sections 2-2-3 and 5-2 (standard reference, not scraped)
A. Myers, P. Colella, and B. Van Straalen, A 4th-Order Particle-in-Cell Method with Phase-Space Remapping for the Vlasov–Poisson Equation, Section 2 (standard reference, not scraped)