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✓ 12 results · all verified · 4 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 8 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

Ewald periodic potentials and particle mesh error

1 · Prerequisites

2 · Summary

Summary

This page defines the neutral mean-zero Poisson problem on an arbitrary full-rank three-dimensional torus and proves the Gaussian Ewald split, its renormalized self term, and uniform real and reciprocal tail bounds. It then defines a causal periodic cardinal-spline assignment with a negative-sign unitary mesh transform. Exact signed aliases, deconvolution normalization, derivative-window bounds, and the $P = 2$ knot branch feed a deterministic pointwise PME potential, energy, and force certificate. The total error keeps real cutoff, reciprocal cutoff, omitted modes, assignment aliases, force differentiation, self correction, and certified arithmetic enclosures as separate terms. Point-source potential claims are made only away from source images.

The evaluator has a separate termination proof covering its explicit integer boxes, finite spline support and self-correction pairs, capped transform subcalls, and bounded enclosure requests. Unresolved cutoffs, arithmetic, branch choices, or tolerances return an explicit status; termination does not require the requested accuracy to be attainable.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Causal cardinal spline window

Definition

For $P = 1$ set $W_{1} (u) = 1_{[0, 1]} (u)$ , including both endpoints. For an integer $P \geq 2$ , define the causal order- $P$ window by $W_{P} (u) = {\begin{cases} \frac{1}{(P - 1)!} \sum_{j = 0}^{P} (- 1)^{j} (\binom{P}{j}) (u - j)_{+}^{P - 1}, & 0 \leq u \leq P, \\ 0, & u < 0 or u > P . \end{cases}$ Here $s_{+} = \max (s, 0)$ , $m!$ is the factorial of the nonnegative integer $m$ with $0! = 1$ , and for integers $0 \leq j \leq P$ , $(\binom{P}{j}) = P! / (j! (P - j)!)$ . Convolution means the integral operation $(f * g) (u) = \int_{R} f (u - s) g (s) d s,$ when the integrand is measurable and integrable, as in Convolution of two functions on $R^{n}$ . The displayed formula is finite at every $u$ and directly defines $W_{P}$ to vanish outside $[0, P]$ ; the local alias lemma proves that it agrees with the $P$ -fold convolution of $W_{1}$ and establishes the resulting integral, positivity, partition, Fourier-transform, and derivative identities.

DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Neutral point sources and Fourier data on a three-dimensional torus

Definition

Let $A \in G L_{3} (R)$ , let $Λ = A Z^{3}$ , let $V = ∣ \det A ∣$ , and let $T_{A} = R^{3} / Λ$ . These are the full-rank lattice and covolume conventions of Full-rank lattices, covolume, and the dual lattice. The reciprocal lattice is $Λ^{*} = A^{- T} Z^{3}$ . For $k \in Z^{3}$ write $κ_{k} = 2 π A^{- T} k, e_{k} (x) = e^{i κ_{k} \cdot x} .$ Thus $e_{k}$ is a character of $T_{A}$ ; the ordinary Fourier frequency is $A^{- T} k$ , while $κ_{k}$ is the angular physical frequency.

A finite point-source configuration consists of an integer $n \geq 0$ , points $x_{1}, \dots, x_{n} \in T_{A}$ that are pairwise distinct when $n > 1$ , and real charges $q_{1}, \dots, q_{n}$ . It is neutral when $\sum_{i = 1}^{n} q_{i} = 0$ . For a smooth periodic test function $φ$ , the point-source distribution is the linear functional $⟨ ρ, φ ⟩ = \sum_{i = 1}^{n} q_{i} φ (x_{i}) .$ Here smooth means that the periodic lift to $R^{3}$ has continuous derivatives of every order, and the test-function topology is generated by the seminorms $\max_{x \in A [0, 1]^{3}} ∣ \partial^{β} φ (x) ∣$ over all derivative multi-indices $β$ . A periodic distribution is a continuous complex-linear functional for this topology. Its distributional Laplacian is defined by $⟨ - Δ u, φ ⟩ = ⟨ u, - Δ φ ⟩$ .

For a smooth periodic $φ$ , its Fourier coefficient is $\hat{φ} (k) = \frac{1}{V} \int_{A (0, 1]^{3}} φ (x) e^{- i κ_{k} \cdot x} d x .$ The source coefficient is $\hat{ρ} (k) = \frac{1}{V} \sum_{i = 1}^{n} q_{i} e^{- i κ_{k} \cdot x_{i}};$ in particular, neutral data have $\hat{ρ} (0) = 0$ . For a periodic distribution $u$ , define $\hat{u} (k) = V^{- 1} ⟨ u, e_{- k} ⟩$ . The mean-zero periodic Poisson solution is a periodic distribution $u$ satisfying $- Δ u = 4 π ρ$ and $\hat{u} (0) = 0$ . This definition fixes the zero-mode convention; no nonneutral periodic solution is implicit.

The Coulomb energy convention is the half-sum of each source charge times the regular part of its potential, after subtracting its own singular term $q_{i} / ∣ x - x_{i} ∣$ . The resulting finite part is called the renormalized self-energy convention. Its Ewald formula is established in this development.

Remarks

The Coulomb normalization is $- Δ u = 4 π ρ$ in three dimensions. No dielectric factor, surface term, nonneutralizing background, or alternative boundary condition is part of this definition.
The dual lattice in the Poisson formula uses exponent $2 π i ξ \cdot x$ ; multiplying its frequency $ξ = A^{- T} k$ by $2 π$ gives $κ_{k}$ .

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Gaussian screened Coulomb kernel and its self limit

Statement

Assume Countable Choice. For $α > 0$ , define $g_{α} (x) = \frac{α^{3}}{π^{3 / 2}} e^{- α^{2} ∣ x ∣^{2}}, \erf (s) = \frac{2}{\sqrt{π}} \int_{0}^{s} e^{- t^{2}} d t, erfc (s) = 1 - \erf (s) .$ Here $Δ = \sum_{j = 1}^{3} \partial_{x_{j}}^{2}$ , $δ_{0}$ is the distribution $⟨ δ_{0}, φ ⟩ = φ (0)$ , and a test function is a smooth compactly supported function $φ$ . For $r = ∣ x ∣ > 0$ put $ϕ_{ℓ} (r) = \erf (α r) / r$ and $ϕ_{s} (r) = erfc (α r) / r$ . Then $ϕ_{ℓ}$ extends continuously to $r = 0$ , $- Δ ϕ_{ℓ} = 4 π g_{α}$ distributionally on $R^{3}$ , and $- Δ ϕ_{s} = 4 π δ_{0} - 4 π g_{α}$ . Moreover, for every $s \geq 0$ , $erfc (s) = \frac{2}{\sqrt{π}} \int_{s}^{\infty} e^{- t^{2}} d t,$ $ϕ_{s} (r) = \frac{1}{r} - \frac{2 α}{\sqrt{π}} + O (r^{2}) (r ↓ 0),$ and for every $r > 0$ , $0 \leq ϕ_{s} (r) \leq \frac{e^{- α^{2} r^{2}}}{\sqrt{π} α r^{2}}, ∣ ϕ_{s}^{'} (r) ∣ \leq \frac{2 α}{\sqrt{π}} \frac{e^{- α^{2} r^{2}}}{r} + \frac{e^{- α^{2} r^{2}}}{\sqrt{π} α r^{3}} .$

Facts & Assumptions

Given: Countable Choice, $α > 0$ , a point $x \in R^{3}$ , $r = ∣ x ∣$ , and an arbitrary $φ \in C_{c}^{\infty} (R^{3})$ .

[A1]

Countable Choice is assumed in the local divergence theorem and its volume interfaces (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The bounded $C^{1}$ divergence theorem applies to punctured balls and smooth vector fields on their closures (Divergence on a bounded C1 Euclidean domain).

[F2]

The integral function of a continuous integrand has derivative equal to that integrand (The first fundamental theorem: if $f$ is integrable on $[a, b]$ and continuous at $c$ , then $F^{'} (c) = f (c)$ ; in particular a continuous $f$ has $F$ as a primitive).

[F3]

$\int_{- \infty}^{\infty} e^{- t^{2}} d t = \sqrt{π}$ (The Gaussian integral $\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt{π}$ ).

[F4]

Absolutely integrable products may be integrated by successive coordinates (Fubini's theorem for L^1 functions on a sigma-finite product).

[F5]

The chain rule differentiates $r \mapsto \erf (α r)$ using its derivative and $r \mapsto α r$ (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ).

[F6]

For $f \in L^{1} (R)$ , the substitution $y = α x$ gives $\int_{R} f (y) d y = α \int_{R} f (α x) d x$ ; its Jacobian is $α > 0$ (A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions).

Proof

technique · direct

1.1givenF2F5algebra

By [F2], $\erf^{'} (s) = 2 e^{- s^{2}} / \sqrt{π}$ , and [F5] gives $h^{'} (r) = 2 α e^{- α^{2} r^{2}} / \sqrt{π}$ for $h (r) = \erf (α r)$ . Differentiating again gives $h^{''} (r) = - 4 α^{3} r e^{- α^{2} r^{2}} / \sqrt{π}$ . Since $∣ e^{- s^{2}} - 1 ∣ = \int_{0}^{s^{2}} e^{- v} d v \leq s^{2}$ , the integral definition yields $h (r) = 2 α r / \sqrt{π} + O (r^{3})$ , and the derivative formula yields $h^{'} (r) = 2 α / \sqrt{π} + O (r^{2})$ . Thus $h (r) / r \to 2 α / \sqrt{π}$ and $(h^{'} (r) r - h (r)) / r^{2} = O (r)$ as $r ↓ 0$ . For $r > 0$ , the radial Laplacian formula $Δ f (∣ x ∣) = (r f (r))^{''} / r$ follows by differentiating $\partial_{x_{j}} f = f^{'} (r) x_{j} / r$ and summing $j = 1, 2, 3$ . Applying it to $f = ϕ_{ℓ} = h / r$ gives $- Δ ϕ_{ℓ} = 4 α^{3} e^{- α^{2} r^{2}} / \sqrt{π} = 4 π g_{α}$ away from the origin.

2.1step 1.1A1F1F3F4F6

Let $F_{ε} = φ \nabla ϕ_{ℓ} - ϕ_{ℓ} \nabla φ$ on the punctured ball $B_{R} ∖ {\overline{B}}_{ε}$ , where $R$ is outside the support of $φ$ . The product rule gives $div F_{ε} = φ Δ ϕ_{ℓ} - ϕ_{ℓ} Δ φ$ . By [F1] its integral is the boundary flux. The outer flux is zero; at the inner sphere both $ϕ_{ℓ}$ and $\nabla ϕ_{ℓ}$ are bounded, and its area is $\int_{0}^{2 π} \int_{0}^{π} ε^{2} \sin θ d θ d ϕ = 4 π ε^{2}$ , so the flux tends to zero. Taking $ε ↓ 0$ proves $\int ϕ_{ℓ} (- Δ φ) = \int φ (- Δ ϕ_{ℓ}) = 4 π \int φ g_{α}$ , using step 1.1. By [F3], [F4], and [F6], applied coordinatewise and with $y = α x$ , $\int_{R^{3}} g_{α} = 1$ .

2.2step 1.1F3algebra

Step 1.1 gives $\erf (α r) = 2 α r / \sqrt{π} + O (r^{3})$ , so $erfc (α r) / r = 1 / r - 2 α / \sqrt{π} + O (r^{2})$ . By [F3] and evenness, $erfc (s) = (2 / \sqrt{π}) \int_{s}^{\infty} e^{- t^{2}} d t \geq 0$ . For $s > 0$ , $\int_{s}^{\infty} e^{- t^{2}} d t \leq s^{- 1} \int_{s}^{\infty} t e^{- t^{2}} d t = e^{- s^{2}} / (2 s)$ ; multiplying by $2 / \sqrt{π}$ gives $erfc (s) \leq e^{- s^{2}} / (\sqrt{π} s)$ . This proves the bound for $ϕ_{s}$ .

3.1A1F1step 2.1algebra

On the punctured ball $Δ (1 / r) = 0$ . Put $G_{ε} = φ \nabla (1 / r) - (1 / r) \nabla φ$ . The product rule gives $div G_{ε} = - (1 / r) Δ φ$ , so [F1] makes the integral of $(1 / r) (- Δ φ)$ equal to its boundary flux. The outer flux is zero. The inner outward normal is $- x / r$ , hence $\partial_{n} (1 / r) = ε^{- 2}$ ; the flux is $\int_{∣ x ∣ = ε} (φ ε^{- 2} - (1 / ε) \partial_{n} φ) d S$ . Since $d S = ε^{2} \sin θ d θ d ϕ$ , its first term tends to $4 π φ (0)$ and the second is bounded by $4 π ε \sup ∣ \nabla φ ∣$ , hence tends to zero. Therefore $- Δ (1 / r) = 4 π δ_{0}$ distributionally. Subtracting step 2.1 proves $- Δ ϕ_{s} = 4 π δ_{0} - 4 π g_{α}$ .

4.1step 2.2algebra∎

Differentiating $ϕ_{s}$ for $r > 0$ gives $ϕ_{s}^{'} (r) = - 2 α e^{- α^{2} r^{2}} / (\sqrt{π} r) - erfc (α r) / r^{2}$ . Apply step 2.2 with $s = α r$ to its second term to obtain the displayed derivative bound.

Remarks

The distributional delta is in the unscreened Coulomb part. The smooth Gaussian term has no delta mass.
The self expansion is the source of the Ewald energy correction $- α \sum_{i} q_{i}^{2} / \sqrt{π}$ after the potential is multiplied by $q_{i} / 2$ .

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Periodic cardinal-spline assignment and unitary mesh transform

Definition

Fix neutral point-source data $(A, V, Λ, κ_{k}, (x_{i}, q_{i})_{i = 1}^{n})$ from Neutral point sources and Fourier data on a three-dimensional torus, an Ewald parameter $α > 0$ , integers $M_{1}, M_{2}, M_{3} \geq 1$ , an assignment order $P \geq 2$ , and write $N_{g} = M_{1} M_{2} M_{3}$ . Let $G = \prod_{r = 1}^{3} {0, \dots, M_{r} - 1}$ and take $W_{P}$ from Causal cardinal spline window. Define the periodic causal assignment weights, for $t \in [0, 1)^{3}$ and $ℓ \in G$ , by $w_{ℓ} (t) = \prod_{r = 1}^{3} \sum_{j \in Z} W_{P} (M_{r} t_{r} - ℓ_{r} - j M_{r}) .$ The sums are finite because $W_{P}$ is supported on $[0, P]$ . They collect every support index into its grid residue; if $P > M_{r}$ , distinct support indices that wrap to the same residue are added. The spread grid is $g_{ℓ} = \sum_{i = 1}^{n} q_{i} w_{ℓ} (t_{i}), t_{i} = A^{- 1} x_{i} (m o d Z^{3}) .$

Use the negative-sign unitary tensor DFT convention ${\hat{g}}_{M} (k) = N_{g}^{- 1 / 2} \sum_{ℓ \in G} g_{ℓ} e^{- 2 π i \sum_{r = 1}^{3} k_{r} ℓ_{r} / M_{r}},$ which is the product transform of The unitary discrete Fourier transform on $Z / N Z$ . Signed frequency representatives are chosen before evaluating windows. A retained set $K \subset Z^{3} ∖ {0}$ is finite, symmetric under $k \mapsto - k$ , injective modulo $(M_{1}, M_{2}, M_{3})$ , and satisfies $∣ k_{r} ∣ < M_{r} / 2$ for each coordinate. In particular, even-grid Nyquist modes are not silently retained.

For $u \in R$ , put $sinc (u) = \sin (u) / u$ when $u \neq 0$ and $sinc (0) = 1$ . Define $D (k) = \prod_{r = 1}^{3} e^{- π i P k_{r} / M_{r}} sinc (π k_{r} / M_{r})^{P} .$ Only modes with $D (k) \neq 0$ are admissible for division. A computed implementation must first certify a strictly positive lower bound for $∣ D (k) ∣$ ; otherwise it returns denominator-inconclusive without dividing. For $k \in K$ , define $a_{k} (t) = e^{- 2 π i k \cdot t}, b_{k} (t) = \frac{\sum_{ℓ \in G} w_{ℓ} (t) e^{- 2 π i \sum_{r} k_{r} ℓ_{r} / M_{r}}}{\overline{D (k)}}, ρ_{k} = \sum_{i} q_{i} a_{k} (t_{i}), z_{k} = \frac{\sqrt{N_{g}} {\hat{g}}_{M} (k)}{\overline{D (k)}} .$ The denominator is the complex conjugate of the printed window multiplier; this sign is part of the definition. Set $c_{k} = \frac{4 π}{V ∣ κ_{k} ∣^{2}} e^{- ∣ κ_{k} ∣^{2} / (4 α^{2})} .$ The exact retained reciprocal energy is $E_{K} = \frac{1}{2} \sum_{k \in K} c_{k} ∣ ρ_{k} ∣^{2},$ and the mesh reciprocal energy is $E_{M} = \frac{1}{2} \sum_{k \in K} c_{k} ∣ z_{k} ∣^{2} .$

Let $d_{k} (t)$ be the declared physical mesh derivative: it is $\nabla_{x} b_{k} (t) = A^{- T} \nabla_{t} b_{k} (t)$ wherever the classical derivative exists. A grid knot is a point with $M_{r} t_{r} \in Z$ for some coordinate $r$ . At a $P = 2$ knot, take the one-sided vector limits from all adjacent open mesh boxes. Define $d_{k} (t)$ to be any selected one-sided limit or the arithmetic average of all those limits. This is a procedure value and does not assert that the classical derivative exists there. The energy-gradient mesh force is $F_{M, i} = - q_{i} \sum_{k \in K} c_{k} Re ⁣ (\overline{z_{k}} d_{k} (t_{i})) .$ Define the diagonal mesh-self correction and corrected reciprocal energy by $C_{M} = \frac{1}{2} \sum_{i} q_{i}^{2} \sum_{k \in K} c_{k} (∣ b_{k} (t_{i}) ∣^{2} - 1), E_{M}^{c o r r} = E_{M} - C_{M} .$ Whether the declared force is the derivative of this corrected energy, and the error bound it satisfies, are proved separately.

Remarks

$\nabla_{x} = A^{- T} \nabla_{t}$ is the physical derivative in the cell coordinates $x = A t$ .
Every reciprocal mode omitted from $K$ , including Nyquist modes, belongs to the separately bounded reciprocal truncation error.
This definition makes no accuracy claim about an FFT or a floating-point implementation. The error theorem supplies those conditions separately.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Poisson periodization of a normalized Gaussian on a full-rank lattice

Statement

Assume Countable Choice and use the full-rank cell and angular frequencies of Neutral point sources and Fourier data on a three-dimensional torus. Here $S (R^{3})$ denotes the Schwartz space of smooth functions whose derivatives decay faster than every inverse polynomial. For $α > 0$ let $g_{α} (x) = \frac{α^{3}}{π^{3 / 2}} e^{- α^{2} ∣ x ∣^{2}} .$ For every $x \in R^{3}$ the two series below converge absolutely, locally uniformly in $x$ , and $\sum_{λ \in Λ} g_{α} (x + λ) = \frac{1}{V} \sum_{k \in Z^{3}} e^{- ∣ κ_{k} ∣^{2} / (4 α^{2})} e^{i κ_{k} \cdot x} .$ The lattice Poisson theorem is applied only to this Schwartz Gaussian; it is not applied to a compactly supported or finite-smoothness assignment window.

Facts & Assumptions

Given: Countable Choice, an invertible cell matrix $A$ , $V = ∣ \det A ∣$ , $Λ = A Z^{3}$ , $Λ^{*} = A^{- T} Z^{3}$ , $κ_{k} = 2 π A^{- T} k$ , $α > 0$ , and $x \in R^{3}$ .

[A1]

Countable Choice is required by the published lattice Poisson theorem and the Gaussian transform supplier (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

For $f \in S (R^{3})$ , both lattice series converge absolutely and the periodization satisfies $\sum_{λ \in Λ} f (x + λ) = V^{- 1} \sum_{ξ \in Λ^{*}} \hat{f} (ξ) e^{2 π i ξ \cdot x}$ under Countable Choice (Poisson summation for a full-rank lattice).

[F2]

With $\hat{f} (ξ) = \int_{R^{3}} f (x) e^{- 2 π i ξ \cdot x} d x$ , $F (e^{- π t ∣ x ∣^{2}}) (ξ) = t^{- 3 / 2} e^{- π ∣ ξ ∣^{2} / t}$ for $t > 0$ under Countable Choice (Euclidean Gaussian transform with the 2π normalization).

[F3]

The normalized Gaussian in the screened-kernel definition is $g_{α} (x) = α^{3} π^{- 3 / 2} e^{- α^{2} ∣ x ∣^{2}}$ (Gaussian screened Coulomb kernel and its self limit).

[F4]

The lattice is $A Z^{3}$ , its dual is $A^{- T} Z^{3}$ , and the physical angular frequency for index $k$ is $κ_{k} = 2 π A^{- T} k$ (Neutral point sources and Fourier data on a three-dimensional torus).

Proof

technique · direct

1.1givenA1F2F3algebra

Put $t = α^{2} / π > 0$ . By [F2], the Fourier transform of $e^{- π t ∣ x ∣^{2}}$ is $t^{- 3 / 2} e^{- π ∣ ξ ∣^{2} / t}$ . Multiplying the input by $α^{3} / π^{3 / 2}$ cancels $t^{- 3 / 2} = (π / α^{2})^{3 / 2}$ , so ${\hat{g}}_{α} (ξ) = e^{- π^{2} ∣ ξ ∣^{2} / α^{2}} .$ Each derivative of $g_{α}$ is a polynomial times $e^{- α^{2} ∣ x ∣^{2}}$ . For every degree $d$ and integer $N$ , choose an integer $m > (d + N) / 2$ ; the inequality $e^{y} \geq y^{m} / m!$ for $y \geq 0$ bounds $∣ x ∣^{d} (1 + ∣ x ∣)^{N} e^{- α^{2} ∣ x ∣^{2}}$ on $∣ x ∣ \geq 1$ , and continuity bounds it on the unit ball. Hence $g_{α} \in S (R^{3})$ , so [F1] applies.

1.2givenF3F4algebra

Fix a compact set $C \subset R^{3}$ and choose $R$ with $∣ x ∣ \leq R$ on $C$ . Since $A$ is invertible, $∣ A n ∣ \geq c ∣ n ∣$ for $c = ∥ A^{- 1} ∥_{2}^{- 1} > 0$ . For $∣ n ∣ \geq 2 R / c$ , $∣ x + A n ∣ \geq c ∣ n ∣ / 2$ uniformly on $C$ , so $g_{α} (x + A n) \leq α^{3} π^{- 3 / 2} e^{- b ∣ n ∣^{2}}$ with $b = α^{2} c^{2} / 4 > 0$ . This majorant is summable: $\sum_{n \in Z^{3}} e^{- b ∣ n ∣^{2}} = (\sum_{j \in Z} e^{- b j^{2}})^{3} < \infty$ , since $e^{- b j^{2}} \leq e^{- b ∣ j ∣}$ for $∣ j ∣ \geq 1$ . The remaining finitely many indices have bounded terms. Bounding every series tail by the corresponding majorant tail proves absolute uniform convergence on $C$ .

2.1A1F1F4step 1.1step 1.2algebra∎

Apply [F1] to $g_{α}$ . Every $ξ \in Λ^{*}$ has the unique form $ξ = A^{- T} k$ with $k \in Z^{3}$ ; by [F4], $2 π ξ = κ_{k}$ . Therefore $π^{2} ∣ ξ ∣^{2} / α^{2} = ∣ κ_{k} ∣^{2} / (4 α^{2})$ and $e^{2 π i ξ \cdot x} = e^{i κ_{k} \cdot x}$ . Substitution in the periodization formula gives exactly the displayed reciprocal series, and [F1] supplies absolute convergence on both sides. At $x = 0$ , absolute convergence of the reciprocal series gives $\sum_{k} e^{- ∣ κ_{k} ∣^{2} / (4 α^{2})} < \infty$ . Since $∣ e^{i κ_{k} \cdot x} ∣ = 1$ , this same sum bounds reciprocal-series tails uniformly in all $x$ . Together with step 1.2 this proves both asserted local uniform convergence claims.

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Uniform lattice-shell counts and explicit Gaussian tail controllers

Statement

Assume Countable Choice. For an invertible real $3 \times 3$ matrix $B$ , put $Γ = B Z^{3}$ and $C_{B} = (2 ∥ B^{- 1} ∥_{2} (1 + \sqrt{3} ∥ B ∥_{2}) + 1)^{3} .$ Here $∥ \cdot ∥_{2}$ is the Euclidean operator norm. Uniformly over every translate $y + Γ$ and every integer shell $m \geq 0$ , $# {z \in y + Γ : m \leq ∣ z ∣ < m + 1} \leq C_{B} (m + 1)^{3} .$

For $β > 0$ , any integer $p \geq 0$ , and integer $M \geq 0$ , set $T_{p} (β, M) = \sum_{m = M}^{\infty} (m + 1)^{p} e^{- β m^{2}} .$ For any integer $J \geq M$ satisfying $2 β J (J + 1) \geq p$ , define $I_{0} (J, β) = \frac{\sqrt{π}}{2 \sqrt{β}} erfc (\sqrt{β} J), I_{1} (J, β) = \frac{e^{- β J^{2}}}{2 β},$ and recursively for $2 \leq s \leq p$ , $I_{s} (J, β) = \frac{J^{s - 1} e^{- β J^{2}}}{2 β} + \frac{s - 1}{2 β} I_{s - 2} (J, β) .$ Then the finite controller $U_{p} (β, M, J) = \sum_{m = M}^{J} (m + 1)^{p} e^{- β m^{2}} + \sum_{s = 0}^{p} (\binom{p}{s}) I_{s} (J, β)$ satisfies $T_{p} (β, M) \leq U_{p} (β, M, J)$ . Such a $J$ exists for every $β > 0$ , $M \geq 0$ , and integer $p \geq 0$ .

Facts & Assumptions

Given: Countable Choice, the full-rank cell matrix $A$ and angular reciprocal lattice from Neutral point sources and Fourier data on a three-dimensional torus, an arbitrary invertible $B \in G L_{3} (R)$ , a translate $y + B Z^{3}$ , $β > 0$ , integer $p \geq 0$ , and $M \in N$ .

[F1]

Each coset of a full-rank lattice has a representative in its closed fundamental parallelotope, whose coordinates lie in $[0, 1]^{3}$ (Fundamental parallelotopes of a lattice tile Euclidean space with covolume volume).

[F2]

$erfc (s)$ is the nonnegative Gaussian tail $\frac{2}{\sqrt{π}} \int_{s}^{\infty} e^{- t^{2}} d t$ for $s \geq 0$ (Gaussian screened Coulomb kernel and its self limit).

[F3]

For every real $X$ there is an integer greater than $X$ (Every complete ordered field is Archimedean).

[F4]

Integration by parts holds on a finite interval when the factors and derivatives satisfy the stated integrability hypotheses (If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ ).

[A1]

Countable Choice is assumed by the Gaussian-tail and lattice-tiling interfaces used here (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

1.1givenA1F1algebra

Under Countable Choice [A1], use [F1] to replace $y$ modulo $B Z^{3}$ by a vector $y_{0} = B v$ with $v \in [0, 1]^{3}$ . Thus $∣ y_{0} ∣ \leq \sqrt{3} ∥ B ∥_{2}$ . If $z = y_{0} + B n$ lies in the shell $m \leq ∣ z ∣ < m + 1$ , then $∣ n ∣ \leq ∥ B^{- 1} ∥_{2} (m + 1 + \sqrt{3} ∥ B ∥_{2})$ . The number of integer triples with this bound is at most $(2 ∥ B^{- 1} ∥_{2} (m + 1 + \sqrt{3} ∥ B ∥_{2}) + 1)^{3} \leq C_{B} (m + 1)^{3}$ , proving the uniform shell count.

1.2givenF3algebra

Choose $J \geq M$ with $2 β J (J + 1) \geq p$ , which exists by [F3]. For $x \geq J$ , the derivative of $f (x) = (x + 1)^{p} e^{- β x^{2}}$ has the sign of $p / (x + 1) - 2 β x \leq 0$ , so $f$ is nonincreasing. Therefore $\sum_{m = J + 1}^{\infty} f (m) \leq \int_{J}^{\infty} f (x) d x$ . Expanding $(x + 1)^{p}$ by the binomial theorem gives $\int_{J}^{\infty} f (x) d x = \sum_{s = 0}^{p} (\binom{p}{s}) I_{s} (J, β)$ .

2.1A1F2F4step 1.2algebra∎

The formulas for $I_{0}$ and $I_{1}$ follow by the substitutions $t = \sqrt{β} x$ and $u = β x^{2}$ . For $s \geq 2$ , choose an integer $L > (s + 1) / 2$ ; $e^{β x^{2}} \geq (β x^{2})^{L} / L!$ shows $x^{s} e^{- β x^{2}} = O (x^{s - 2 L})$ is integrable at infinity. Apply [F4] on $[J, R]$ and let $R \to \infty$ ; the same inequality makes the boundary term $R^{s - 1} e^{- β R^{2}}$ vanish. This gives $I_{s} = J^{s - 1} e^{- β J^{2}} / (2 β) + (s - 1) I_{s - 2} / (2 β)$ . Hence the finite sum through $J$ plus the integral tail in step 1.2 equals $U_{p}$ and bounds $T_{p}$ .

Remarks

The shell estimate is intentionally coarse. Its cubic factor is enough because every later term has Gaussian decay; no asymptotic lattice-point theorem is used.
For real-space estimates take $B = A$ . For reciprocal estimates take $B = 2 π A^{- T}$ , whose lattice points are exactly the angular physical frequencies $κ_{k}$ .

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Smooth torus test functions have rapidly convergent Fourier pairings

Statement

Assume Countable Choice. Let $A \in G L_{3} (R)$ , $V = ∣ \det A ∣$ , $Λ = A Z^{3}$ , $κ_{k} = 2 π A^{- T} k$ , and let $φ$ be a smooth $Λ$ -periodic function. With the Fourier coefficients of Neutral point sources and Fourier data on a three-dimensional torus, for every derivative order $q \geq 0$ , $\sum_{k \in Z^{3}} ∣ κ_{k} ∣^{q} ∣ \hat{φ} (k) ∣ < \infty .$ Consequently the Fourier series of $φ$ and every differentiated series converge uniformly to $φ$ and its corresponding derivatives. More generally, if a coefficient sequence satisfies $∣ a_{k} ∣ \leq C (1 + ∣ κ_{k} ∣)^{s}$ for some $C < \infty$ and $s \geq 0$ , then $T_{a} (φ) : = V \sum_{k \in Z^{3}} a_{k} \hat{φ} (- k)$ converges absolutely and defines a continuous periodic distribution. A periodic distribution is determined by its Fourier coefficients.

Facts & Assumptions

Given: Countable Choice, $A \in G L_{3} (R)$ , the lattice data of Neutral point sources and Fourier data on a three-dimensional torus, a smooth periodic $φ$ , and a sequence $(a_{k})_{k \in Z^{3}}$ with $∣ a_{k} ∣ \leq C (1 + ∣ κ_{k} ∣)^{s}$ for fixed finite $C$ and $s \geq 0$ .

[A1]

Countable Choice is assumed by the periodic Fourier uniqueness theorem used below (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The cell has volume $V$ , its angular characters are $e_{k} (x) = e^{i κ_{k} \cdot x}$ , and the Fourier coefficient is $V^{- 1} \int_{F} φ (x) e^{- i κ_{k} \cdot x} d x$ (Neutral point sources and Fourier data on a three-dimensional torus, Fundamental parallelotopes of a lattice tile Euclidean space with covolume volume).

[F2]

The lattice characters are orthogonal on $F$ : $V^{- 1} \int_{F} e^{i (κ_{k} - κ_{j}) \cdot x} d x = 1$ for $k = j$ and $0$ otherwise (Orthogonality of the lattice characters over a fundamental domain).

[F3]

Integration by parts on a closed interval holds for differentiable factors with integrable derivatives; periodic endpoint values cancel when the factors are periodic (If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ ).

[F4]

For every real $p > 1$ , $\sum_{m \geq 1} m^{- p}$ converges (The p-series for a real exponent p converges exactly when p is greater than one).

[F5]

A function series dominated termwise by a convergent scalar series converges uniformly (Weierstrass M-test for complex-valued function series).

[F6]

Uniform convergence of the derivatives, together with convergence at one point, identifies the derivative of the uniform limit on a closed interval (If continuously differentiable functions converge at one point and their derivatives converge uniformly on a closed interval, then the functions converge uniformly to a differentiable function whose derivative is the derivative limit).

[F7]

A continuous periodic function with all Fourier coefficients zero vanishes identically on the torus (Continuous lattice-periodic functions are determined by their lattice Fourier coefficients).

[F8]

Linear change of variables sends $F$ to the fundamental cube, and an absolutely integrable function on the cube may be integrated successively in its three coordinates (A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions, Fubini's theorem for L^1 functions on a sigma-finite product).

Proof

technique · direct

1.1A1F1F3F8algebra

Put $\tilde{φ} (t) = φ (A t)$ , which is smooth and $Z^{3}$ -periodic. Change variables to the unit cube and integrate successively in the three coordinates using [F8]. In $t$ coordinates the physical Laplacian is the constant-coefficient operator $Δ_{t} = \sum_{r, s} (A^{- 1} A^{- T})_{r s} \partial_{t_{r}} \partial_{t_{s}}$ . Repeatedly apply [F3] to the real and imaginary parts in each coordinate; periodicity makes every opposite-face boundary term cancel, and the character is an eigenfunction with eigenvalue $- ∣ κ_{k} ∣^{2}$ . Thus for every integer $m \geq 1$ , $\hat{φ} (k) = (1 + ∣ κ_{k} ∣^{2})^{- m} \hat{(1 - Δ)^{m} φ} (k), ∣ \hat{φ} (k) ∣ \leq (1 + ∣ κ_{k} ∣^{2})^{- m} \sup_{x \in F} ∣ (1 - Δ)^{m} φ (x) ∣ .$ The last inequality follows because the Fourier coefficient is the normalized integral over the cell [F1].

2.1step 1.1F4algebra

Since $∣ κ_{k} ∣ \geq 2 π ∣ k ∣ / ∥ A^{T} ∥_{2}$ , the number of integer vectors with $j \leq ∣ κ_{k} ∣ < j + 1$ is at most $(2 ∥ A^{T} ∥_{2} (j + 1) / (2 π) + 3)^{3}$ . For any fixed $q$ , choose an integer $m > (q + 4) / 2$ ; the shell sum of $∣ κ_{k} ∣^{q} (1 + ∣ κ_{k} ∣^{2})^{- m}$ is then bounded by a constant times a convergent p-series [F4]. Hence $\sum_{k} ∣ κ_{k} ∣^{q} ∣ \hat{φ} (k) ∣ < \infty$ .

3.1step 2.1F2F3F5F6F7construct

Enumerate $Z^{3}$ by increasing $\max_{r} ∣ k_{r} ∣$ and lexicographically within each finite shell. For each derivative order, step 2.1 and [F5] make the Fourier series and its formal derivative series uniformly convergent. Apply [F6] to the real and imaginary parts along each coordinate line; the uniform derivative limits are the derivatives of the sum. By [F2], termwise integration shows that the sum has the same Fourier coefficients as $φ$ ; [F7] makes the continuous difference zero. Each derivative series likewise has the coefficients of the corresponding derivative of $φ$ , by periodic integration by parts [F3], so [F7] identifies it with that derivative. Hence the Fourier partial sums converge in every smooth seminorm.

4.1step 1.1step 2.1A1F1F2algebra∎

Choose $m > (s + 4) / 2$ in step 1.1. The shell count of step 2.1 gives $\sum_{k} ∣ a_{k} \hat{φ} (- k) ∣ \leq C \sup_{F} ∣ (1 - Δ)^{m} φ ∣ \sum_{k} (1 + ∣ κ_{k} ∣)^{s} (1 + ∣ κ_{k} ∣^{2})^{- m} < \infty .$ This is a bound by the seminorm $\sup_{F} ∣ (1 - Δ)^{m} φ ∣$ , a finite sum of the smooth seminorms in the source-data definition, so $T_{a}$ is a continuous linear functional. If a periodic distribution has the same Fourier coefficients as $T_{a}$ , their difference vanishes on every character and hence on every trigonometric polynomial; by the smooth convergence of step 3.1 and continuity of the distribution, it vanishes on every smooth test function. Therefore the coefficients determine it.

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Exact three-dimensional spline assignment aliases and FFT normalization

Statement

Use the periodic cardinal-assignment model of Periodic cardinal-spline assignment and unitary mesh transform, with integers $M_{r} \geq 1$ , order $P \geq 2$ , grid size $N_{g} = \prod_{r} M_{r}$ , a retained signed mode $k$ satisfying $∣ k_{r} ∣ < M_{r} / 2$ , and $D (k) \neq 0$ . Put $u_{r} = k_{r} / M_{r}$ , $a_{k} (t) = e^{- 2 π i k \cdot t}$ , and $r_{k, n} = \prod_{r = 1}^{3} \frac{\overline{D_{r} (k_{r} + n_{r} M_{r})}}{\overline{D_{r} (k_{r})}}, D_{r} (m) = e^{- π i P m / M_{r}} sinc (π m / M_{r})^{P} .$ Then the single-particle assignment factor and the spread-grid coefficient obey the absolutely convergent identities $b_{k} (t) = \sum_{n \in Z^{3}} r_{k, n} a_{k + n ⊙ M} (t), z_{k} = \sum_{n \in Z^{3}} r_{k, n} ρ_{k + n ⊙ M},$ where $(n ⊙ M)_{r} = n_{r} M_{r}$ and $ρ_{j} = \sum_{i} q_{i} e^{- 2 π i j \cdot t_{i}}$ . In particular $z_{k} - ρ_{k} = \sum_{n \neq 0} r_{k, n} ρ_{k + n ⊙ M} .$ The mesh coefficient also has the exact finite representation $z_{k} = \sum_{i} q_{i} b_{k} (t_{i})$ , so no cell-volume or grid-size factor remains after the displayed $\sqrt{N_{g}}$ conversion. The alias bound below uses only the Fourier series of the compact spline; the lattice Poisson theorem is not applied to that spline.

For $u_{r} \neq 0$ define $a_{r} (P, u_{r}) = \sum_{m \neq 0} {∣ \frac{u_{r}}{u_{r} + m} ∣}^{P}, a_{r} (P, 0) = 0, A_{k} = \prod_{r = 1}^{3} (1 + a_{r} (P, u_{r})) - 1.$ Then $∣ b_{k} (t) - a_{k} (t) ∣ \leq A_{k}$ , $∣ b_{k} (t) ∣ \leq 1 + A_{k}$ , and $∣ z_{k} - ρ_{k} ∣ \leq Q A_{k}$ for $Q = \sum_{i} ∣ q_{i} ∣$ . For $J \geq 1$ and $0 < ∣ u_{r} ∣ < 1 / 2$ , $a_{r} (P, u_{r}) \leq 2 \frac{∣ u_{r} ∣^{P}}{(1 - ∣ u_{r} ∣)^{P}} \sum_{m = 1}^{\infty} m^{- P}, \sum_{∣ m ∣ > J} {∣ \frac{u_{r}}{u_{r} + m} ∣}^{P} \leq 2 \frac{∣ u_{r} ∣^{P}}{(1 - ∣ u_{r} ∣)^{P}} \frac{⌊ J ⌋^{1 - P}}{P - 1} .$ Thus every alias controller has a finite p-series remainder bound. The periodic assignment also satisfies $\sum_{ℓ \in G} w_{ℓ} (t) = 1$ , so the spread-grid zero mode is $\sum_{ℓ} g_{ℓ} = \sum_{i} q_{i} = 0$ for neutral data.

Facts & Assumptions

Given: A full-rank cell, neutral finite charges, integers $M_{r} \geq 1$ , $P \geq 2$ , $k \in Z^{3}$ with $∣ k_{r} ∣ < M_{r} / 2$ , and the periodic assignment, DFT, $D_{r}$ , $b_{k}$ , $z_{k}$ , and $ρ_{k}$ definitions of Neutral point sources and Fourier data on a three-dimensional torus and Periodic cardinal-spline assignment and unitary mesh transform.

[F1]

$W_{1} = 1_{[0, 1]}$ and, for $P \geq 2$ , $W_{P}$ is the finite truncated-power formula supported on $[0, P]$ ; convolution is the integral operation defined in Convolution of two functions on $R^{n}$ . The periodic weights, signed mode restrictions, and denominator $D (k) = \prod_{r} D_{r} (k_{r})$ have the definitions displayed on the A page (Causal cardinal spline window, Periodic cardinal-spline assignment and unitary mesh transform).

[F2]

The mesh transform has the negative-sign unitary normalization $N_{g}^{- 1 / 2}$ in dimension three (Periodic cardinal-spline assignment and unitary mesh transform, The unitary discrete Fourier transform on $Z / N Z$ ).

[F3]

For every real $p > 1$ , the p-series $\sum_{m \geq 1} m^{- p}$ converges (The p-series for a real exponent p converges exactly when p is greater than one).

[F4]

Nonnegative iterated integrals may be interchanged; this justifies the convolution and periodic partition calculations (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F5]

An absolutely summable Fourier coefficient series converges uniformly by the complex Weierstrass test (Weierstrass M-test for complex-valued function series).

[F6]

The derivatives of sine and cosine are cosine and minus sine, so the real and imaginary parts of $e^{i ω x}$ have the elementary primitives used to evaluate its finite-interval integral (The derivatives of sine and cosine are cosine and minus sine).

[F7]

Absolutely integrable products may be interchanged and factored in the convolution transform calculation (Fubini's theorem for L^1 functions on a sigma-finite product).

Proof

technique · direct

1.1F1algebra

Define $V_{1} = 1_{[0, 1]}$ and recursively $V_{P} (u) = \int_{0}^{1} V_{P - 1} (u - s) d s$ , so $V_{P} = V_{P - 1} * 1_{[0, 1]}$ by the convolution definition [F1]. For $P = 2$ , the integral is the length of $[0, 1] \cap [u - 1, u]$ , giving $u$ on $[0, 1]$ , $2 - u$ on $[1, 2]$ , and zero outside; this is the truncated-power formula. For $P \geq 3$ , integrate the formula for $V_{P - 1}$ term by term; reindexing the second sum and applying Pascal's identity gives the stated formula for $W_{P}$ . Thus $V_{P} = W_{P}$ for all $P \geq 2$ . The recursion gives nonnegativity and support in $[0, P]$ . The formula gives $W_{P} (0) = 0$ , and its value at $P$ is the $P$ th finite difference of the degree- $(P - 1)$ polynomial $u^{P - 1}$ , hence is zero; therefore the zero extension is continuous at both support endpoints. Differentiating the formula on each polynomial interval and applying Pascal's identity gives $W_{P}^{'} (u) = W_{P - 1} (u) - W_{P - 1} (u - 1)$ ; for $P = 2$ this identity holds on open intervals, with the one-sided values at integers.

1.2givenF3algebra

From the transform factor, for $n_{r} \neq 0$ and $u_{r} = k_{r} / M_{r}$ , $∣ D_{r} (k_{r} + n_{r} M_{r}) / D_{r} (k_{r}) ∣ = ∣ u_{r} / (u_{r} + n_{r}) ∣^{P}$ . The phase factors have modulus one; if $u_{r} = 0$ , every shifted sinc numerator vanishes. Since $∣ u_{r} ∣ < 1 / 2$ , $∣ u_{r} + n ∣ \geq (1 - ∣ u_{r} ∣) ∣ n ∣$ . Summing over positive and negative $n$ and applying [F3] gives the total bound for $a_{r}$ . For real $J \geq 1$ put $L = ⌊ J ⌋ \geq 1$ . Since $x^{- P}$ decreases, $n^{- P} \leq \int_{n - 1}^{n} x^{- P} d x$ for each integer $n > L$ . Summing and taking the limit gives $\sum_{n > J} n^{- P} \leq \int_{L}^{\infty} x^{- P} d x = L^{1 - P} / (P - 1)$ , proving the stated finite remainder, including noninteger thresholds. The tensor product has total nonzero alias mass $A_{k} = \prod_{r} (1 + a_{r}) - 1$ .

2.1F1F6F7step 1.1algebra

From step 1.1, $W_{P}$ is the convolution of $P$ copies of $1_{[0, 1]}$ . Fubini [F7] factors its Fourier integral as ${\hat{W}}_{P} (ω) = (\int_{0}^{1} e^{- i ω s} d s)^{P}$ . For $ω \neq 0$ , the sine and cosine primitives from [F6] give $\int_{0}^{1} e^{- i ω s} d s = e^{- i ω / 2} sinc (ω / 2)$ ; at $ω = 0$ the integral is $1$ . Therefore ${\hat{W}}_{P} (ω) = e^{- i P ω / 2} sinc (ω / 2)^{P}$ and $\int W_{P} = 1$ . The same primitive gives $\int_{0}^{1} e^{2 π i m v} d v = 1$ for $m = 0$ and $0$ for nonzero integers $m$ .

2.2F1F4step 1.1algebra

For $P = 2$ , write $W_{2} (a - ℓ) = \int_{0}^{1} W_{1} (a - ℓ - s) d s$ and interchange the nonnegative sum and integral by [F4]. Except at finitely many $s$ for which $a - s$ is an integer, exactly one integer $ℓ$ has $a - ℓ - s \in [0, 1]$ ; the exceptional set has integral zero, so $\sum_{ℓ \in Z} W_{2} (a - ℓ) = 1$ . For $P > 2$ , use $W_{P} (x) = \int_{0}^{1} W_{P - 1} (x - s) d s$ and Tonelli [F4]; the inductive identity for $P - 1$ makes the sum inside the integral equal to $1$ . Thus $\sum_{ℓ \in Z} W_{P} (a - ℓ) = 1$ for every $P \geq 2$ .

3.1F1F3F5step 2.1algebra

For one coordinate, set $θ = 2 π k / M$ and $H (a) = e^{i θ a} \sum_{ℓ \in Z} W_{P} (a - ℓ) e^{- i θ ℓ}$ . The sum is locally finite and continuous, and reindexing shows $H (a + 1) = H (a)$ . Only finitely many $ℓ$ contribute on $[0, 1]$ . Substitution $s = a - ℓ$ and partition of $R$ into $[- ℓ, 1 - ℓ]$ give its $m$ th Fourier coefficient as $\int_{R} W_{P} (s) e^{- i (2 π m - θ) s} d s = {\hat{W}}_{P} (2 π m - θ)$ , using step 2.1. For $∣ k ∣ < M / 2$ , these coefficients are $O ((1 + ∣ m ∣)^{- P})$ , hence absolutely summable by [F3]. Their Fourier series converges uniformly by [F5] to a continuous function with the same coefficients as $H$ .

3.2givenF1step 2.2algebra

In one coordinate, summing the periodic residue weights over $ℓ = 0, \dots, M_{r} - 1$ is the same as summing $W_{P} (M_{r} t_{r} - L)$ over all integers $L = ℓ + j M_{r}$ ; by step 2.2 this sum is $1$ . Taking the product over the three coordinates gives $\sum_{ℓ \in G} w_{ℓ} (t) = 1$ . Therefore $\sum_{ℓ} g_{ℓ} = \sum_{i} q_{i} \sum_{ℓ} w_{ℓ} (t_{i}) = \sum_{i} q_{i} = 0$ for neutral data.

4.1step 2.1step 3.1algebra

Let $h$ be the difference between $H$ and the uniform Fourier sum in step 3.1. Every Fourier coefficient of $h$ is zero. For $J \geq 1$ , define $K_{J} (v) = J^{- 1} ∣ \sum_{r = 0}^{J - 1} e^{2 π i r v} ∣^{2}$ . It is a finite nonnegative trigonometric polynomial with integral $1$ by step 2.1; the vanishing coefficients of $h$ therefore give $K_{J} * h = 0$ . Given $ε > 0$ , uniform continuity of $h$ gives $δ \in (0, 1 / 2)$ such that $∣ h (a - v) - h (a) ∣ < ε$ when $v$ is within $δ$ of an integer. Away from those points, the geometric-sum formula gives $K_{J} (v) \leq [J \sin^{2} (π δ)]^{- 1}$ . Hence $∥ K_{J} * h - h ∥_{\infty} \leq ε + 2 ∥ h ∥_{\infty} / (J \sin^{2} (π δ))$ . Letting $J \to \infty$ and then $ε ↓ 0$ yields $h = 0$ . Thus the series in step 3.1 equals $H$ .

5.1F1step 2.1step 3.1step 4.1algebra

Multiply the identity $H (a) = \sum_{m} {\hat{W}}_{P} (2 π m - θ) e^{2 π i m a}$ established by steps 3.1 and 4.1 by $e^{- i θ a}$ , put $a = M t$ , and reindex $n = - m$ . Since $W_{P}$ is real, ${\hat{W}}_{P} (- ω) = \overline{{\hat{W}}_{P} (ω)}$ ; step 2.1 then gives $\sum_{ℓ \in Z} W_{P} (M t - ℓ) e^{- 2 π i k ℓ / M} = \sum_{n \in Z} \overline{D_{r} (k + n M)} e^{- 2 π i (k + n M) t}$ . In the finite-grid periodic sum, each index $ℓ + j M$ has the same phase as residue $ℓ$ , since $e^{- 2 π i k j} = 1$ . Thus the identity holds for the periodized assignment weights.

6.1givenF2step 5.1algebra

Apply step 5.1 in each of the three coordinates and multiply the absolutely convergent series. Dividing by $\overline{D (k)} = \prod_{r} \overline{D_{r} (k_{r})}$ yields $b_{k} (t) = \sum_{n \in Z^{3}} r_{k, n} e^{- 2 π i (k + n ⊙ M) \cdot t}$ . By [F2], $\sqrt{N_{g}} {\hat{g}}_{M} (k) = \sum_{i} q_{i} \sum_{ℓ} w_{ℓ} (t_{i}) e^{- 2 π i k \cdot ℓ / M}$ . Substitute the single-particle identity, divide by $\overline{D (k)}$ , and interchange the absolutely convergent alias series with the finite particle sum. This gives $z_{k} = \sum_{n} r_{k, n} ρ_{k + n ⊙ M}$ with exactly the stated $\sqrt{N_{g}}$ factor and no cell-volume factor.

7.1step 6.1step 1.2algebra∎

Since $∣ a_{k} (t) ∣ = 1$ , the identity for $b_{k}$ gives $∣ b_{k} - a_{k} ∣ \leq A_{k}$ and $∣ b_{k} ∣ \leq 1 + A_{k}$ . Also $∣ ρ_{j} ∣ \leq \sum_{i} ∣ q_{i} ∣ = Q$ for every $j$ , so the exact identity for $z_{k}$ gives $∣ z_{k} - ρ_{k} ∣ \leq Q A_{k}$ . These are the asserted source-assignment alias bounds.

Remarks

The negative-sign unitary transform produces the conjugate multiplier $\overline{D (k)}$ for the causal window $W_{P} (M t - ℓ)$ . Dividing by $D (k)$ would introduce a deterministic phase error.
Periodic wrapping accumulates all predecessor support residues, including duplicate residues when $P > M_{r}$ ; the proof uses that full finite sum.
The proof of the spline Fourier series is local and choice-free. The Countable Choice branch enters later through Gaussian Poisson summation.

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The mean-zero Fourier distribution solves the neutral periodic Poisson equation

Statement

Assume Countable Choice and a neutral finite point-source configuration on the cell $A Z^{3}$ of Neutral point sources and Fourier data on a three-dimensional torus. Put $\hat{u} (0) = 0, \hat{u} (k) = \frac{4 π}{V ∣ κ_{k} ∣^{2}} \sum_{i = 1}^{n} q_{i} e^{- i κ_{k} \cdot x_{i}} (k \neq 0) .$ The polynomially bounded sequence $\hat{u} (k)$ defines a periodic distribution $u$ by Fourier pairing. It is the unique mean-zero periodic distribution satisfying $- Δ u = 4 π ρ$ .

Facts & Assumptions

Given: Countable Choice, $A \in G L_{3} (R)$ , a finite configuration $(x_{i}, q_{i})_{i = 1}^{n}$ with $\sum_{i} q_{i} = 0$ , and a smooth periodic test function $φ$ .

[A1]

The proof uses Countable Choice through the smooth Fourier pairing and the lattice periodic uniqueness supplier (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The source distribution satisfies $⟨ ρ, φ ⟩ = \sum_{i} q_{i} φ (x_{i})$ and its Fourier coefficient is $V^{- 1} \sum_{i} q_{i} e^{- i κ_{k} \cdot x_{i}}$ ; distributions are defined by continuous linear pairings on smooth periodic tests (Neutral point sources and Fourier data on a three-dimensional torus).

[F2]

Every polynomially bounded coefficient sequence defines a continuous periodic distribution by $V \sum_{k} a_{k} \hat{φ} (- k)$ ; smooth Fourier series converge to the test function in every smooth seminorm (Smooth torus test functions have rapidly convergent Fourier pairings).

[F3]

Since $e_{k} (x) = e^{i κ_{k} \cdot x}$ , one has $- Δ e_{k} = ∣ κ_{k} ∣^{2} e_{k}$ and the zero character is the constant function $1$ (Neutral point sources and Fourier data on a three-dimensional torus).

Proof

technique · direct

1.1givenA1F1F2algebra

Define $\hat{u}$ by the displayed coefficients. For $k \neq 0$ , $∣ \hat{u} (k) ∣ \leq \frac{4 π Q}{V ∣ κ_{k} ∣^{2}}, Q = \sum_{i} ∣ q_{i} ∣,$ so the sequence is polynomially bounded. Under Countable Choice [A1], [F2] therefore defines a continuous periodic distribution $u$ with exactly these coefficients; its zero Fourier coefficient is zero by construction.

1.2givenA1F1F2F3algebra

For a smooth periodic $φ$ , use Countable Choice [A1], [F3], and the coefficient pairing of [F2] to compute $⟨ u, - Δ φ ⟩ = V \sum_{k \neq 0} \hat{u} (k) ∣ κ_{k} ∣^{2} \hat{φ} (- k) = 4 π \sum_{i} q_{i} \sum_{k \neq 0} e^{- i κ_{k} \cdot x_{i}} \hat{φ} (- k) .$ The sums converge absolutely by [F2]. Reindex the full Fourier series by $j = - k$ ; [F2] gives $\sum_{k \in Z^{3}} e^{- i κ_{k} \cdot x_{i}} \hat{φ} (- k) = φ (x_{i})$ . The omitted zero term contributes $\hat{φ} (0) \sum_{i} q_{i} = 0$ by neutrality. Hence $⟨ u, - Δ φ ⟩ = 4 π \sum_{i} q_{i} φ (x_{i}) = 4 π ⟨ ρ, φ ⟩$ , proving $- Δ u = 4 π ρ$ .

2.1A1F2F3algebra∎

If $u_{1}, u_{2}$ are two mean-zero periodic distribution solutions, let $h = u_{1} - u_{2}$ . Testing $- Δ h = 0$ against each character gives $∣ κ_{k} ∣^{2} \hat{h} (k) = 0$ for every $k \neq 0$ . Thus all nonzero Fourier coefficients of $h$ vanish, and its zero coefficient vanishes because both solutions have mean zero. Under [A1], the smooth Fourier partial sums converge in every smooth seminorm by [F2]; continuity of $h$ as a distribution then makes $h$ vanish on every smooth test function. Therefore $u_{1} = u_{2}$ .

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Derivative-window alias bounds including the P=2 knot branch

Statement

Use the data and aliases of Exact three-dimensional spline assignment aliases and FFT normalization. Write $h_{r} (0) = 1$ and, for $u_{r} = k_{r} / M_{r} \neq 0$ , $h_{r} (n) = ∣ u_{r} / (u_{r} + n) ∣^{P}$ for $n \neq 0$ ; when $u_{r} = 0$ , set $h_{r} (n) = 0$ for $n \neq 0$ . Put $a_{r} = \sum_{n \neq 0} h_{r} (n) .$ For $P \geq 3$ , define $v_{r} = \sum_{n \neq 0} ∣ k_{r} + n M_{r} ∣ h_{r} (n), J_{k, r} = (∣ k_{r} ∣ + v_{r}) \prod_{s \neq r} (1 + a_{s}) - ∣ k_{r} ∣,$ and $B_{k} = 2 π ∥ A^{- T} ∥_{2} {(\sum_{r = 1}^{3} J_{k, r}^{2})}^{1 / 2} .$ For $P \geq 3$ the physical derivative alias error satisfies $∥ \nabla_{x} b_{k} (t) - \nabla_{x} a_{k} (t) ∥_{2} \leq B_{k} .$ Each series has a finite p-series controller: for $0 < ∣ u_{r} ∣ < 1 / 2$ , $a_{r} \leq 2 \frac{∣ u_{r} ∣^{P}}{(1 - ∣ u_{r} ∣)^{P}} ζ (P), v_{r} \leq 2 M_{r} \frac{∣ u_{r} ∣^{P}}{(1 - ∣ u_{r} ∣)^{P - 1}} ζ (P - 1),$ where $ζ (s) = \sum_{m \geq 1} m^{- s}$ for $s > 1$ . For $J_{0} \geq 1$ , after summing exactly over $1 \leq ∣ n ∣ \leq J_{0}$ , the respective omitted tails are at most $2 \frac{∣ u_{r} ∣^{P}}{(1 - ∣ u_{r} ∣)^{P}} \frac{⌊ J_{0} ⌋^{1 - P}}{P - 1}, 2 M_{r} \frac{∣ u_{r} ∣^{P}}{(1 - ∣ u_{r} ∣)^{P - 1}} \frac{⌊ J_{0} ⌋^{2 - P}}{P - 2} .$

For the approved $P = 2$ case, the absolute derivative p-series above diverges when $u_{r} \neq 0$ . Instead use the finite derivative of the local piecewise window sum. At every open mesh box and each one-sided knot limit, put $C_{k, r} = \frac{2 M_{r}}{∣ D (k) ∣} + 2 π ∣ k_{r} ∣, B_{k} = ∥ A^{- T} ∥_{2} {(\sum_{r = 1}^{3} C_{k, r}^{2})}^{1 / 2} .$ This is a computable finite bound for the same physical derivative error. At a grid knot, use the declared $d_{k} (t)$ branch from Periodic cardinal-spline assignment and unitary mesh transform: it is a selected one-sided value or their average and obeys the bound, but is not asserted to be a classical derivative there. Thus for the full $P \geq 2$ range the bound is on $d_{k} (t) - \nabla_{x} a_{k} (t)$ , with $d_{k} = \nabla_{x} b_{k}$ wherever the classical derivative exists.

Facts & Assumptions

Given: A cell matrix $A$ , integers $M_{r} \geq 1$ , spline order $P \geq 2$ , signed mode $k$ with $∣ k_{r} ∣ < M_{r} / 2$ and $D (k) \neq 0$ , a fractional point $t \in [0, 1)^{3}$ , and the assignment factors $a_{k}, b_{k}$ of the periodic mesh definition.

[F1]

The exact signed alias expansion of $b_{k}$ has coefficients $r_{k, n}$ with $∣ r_{k, n} ∣ = \prod_{r} h_{r} (n_{r})$ , and $W_{P}$ is the causal $P$ -fold cardinal window (Exact three-dimensional spline assignment aliases and FFT normalization, Periodic cardinal-spline assignment and unitary mesh transform).

[F2]

The physical gradient is $\nabla_{x} = A^{- T} \nabla_{t}$ , and $\nabla_{x} a_{k} = - 2 π i A^{- T} k a_{k}$ (Neutral point sources and Fourier data on a three-dimensional torus).

[F3]

For $s > 1$ , $ζ (s) = \sum_{m \geq 1} m^{- s}$ converges (The p-series for a real exponent p converges exactly when p is greater than one).

[F4]

Uniform convergence of a series and its derivative series on a closed interval permits termwise differentiation, with real and imaginary parts treated separately (If continuously differentiable functions converge at one point and their derivatives converge uniformly on a closed interval, then the functions converge uniformly to a differentiable function whose derivative is the derivative limit).

Proof

technique · direct

1.1givenF3algebra

Suppose $P \geq 3$ and first take $u_{r} \neq 0$ . For $n \neq 0$ , $h_{r} (n) \leq \frac{∣ u_{r} ∣^{P}}{(1 - ∣ u_{r} ∣)^{P} ∣ n ∣^{P}}, ∣ k_{r} + n M_{r} ∣ h_{r} (n) \leq M_{r} \frac{∣ u_{r} ∣^{P}}{(1 - ∣ u_{r} ∣)^{P - 1} ∣ n ∣^{P - 1}},$ because $∣ u_{r} + n ∣ \geq (1 - ∣ u_{r} ∣) ∣ n ∣$ . Since $P > 1$ and $P - 1 > 1$ , [F3] proves absolute convergence and the stated $a_{r}, v_{r}$ bounds. To prove the remainders locally, put $L = ⌊ J_{0} ⌋ \geq 1$ . For $s > 1$ and each integer $m > L$ , monotonicity gives $m^{- s} \leq \int_{m - 1}^{m} x^{- s} d x$ . Summing and taking the limit yields $\sum_{m > J_{0}} m^{- s} \leq \int_{L}^{\infty} x^{- s} d x = L^{1 - s} / (s - 1)$ . Apply this with $s = P$ and $s = P - 1$ to the two majorants above. If $u_{r} = 0$ , all shifted factors and both series vanish.

1.2F1F2algebra

Now let $P = 2$ . The convolution $W_{2} (u) = \int_{0}^{1} 1_{[0, 1]} (u - s) d s$ is the length of $[0, 1] \cap [u - 1, u]$ , hence is $u$ on $[0, 1]$ , $2 - u$ on $[1, 2]$ , and $0$ outside $[0, 2]$ . Its derivative is $1$ on $(0, 1)$ , $- 1$ on $(1, 2)$ , and $0$ outside, with the indicated one-sided values at $0, 1, 2$ . On every open mesh box, only finitely many window translates are nonzero, so direct differentiation gives $\partial_{t_{r}} b_{k} (t) = \frac{M_{r}}{\overline{D (k)}} \sum_{ℓ \in Z^{3}} W_{2}^{'} (M_{r} t_{r} - ℓ_{r}) \prod_{s \neq r} W_{2} (M_{s} t_{s} - ℓ_{s}) e^{- 2 π i \sum_{s} k_{s} ℓ_{s} / M_{s}} .$ The periodic residue version is identical because $k_{s}$ are integers and the phase is unchanged when $ℓ_{s}$ is shifted by $M_{s}$ . For every non-knot $y$ , one translate lies in $(0, 1)$ and one in $(1, 2)$ , so $\sum_{ℓ_{r}} ∣ W_{2}^{'} (y - ℓ_{r}) ∣ = 2$ ; at a knot the same sum of absolute one-sided values is at most $2$ . Also $W_{2} \geq 0$ and $\sum_{ℓ} W_{2} (y - ℓ) = 1$ by the partition identity proved in [F1]. Therefore $∣ \partial_{t_{r}} b_{k} ∣ \leq 2 M_{r} / ∣ D (k) ∣$ . Since $∣ \partial_{t_{r}} a_{k} ∣ = 2 π ∣ k_{r} ∣$ , the fractional derivative error is at most $C_{k, r}$ .

2.1F1F4step 1.1algebra

For $P \geq 3$ , [F1] gives $b_{k} (t) = \sum_{n \in Z^{3}} r_{k, n} e^{- 2 π i (k + n ⊙ M) \cdot t} .$ The absolute derivative sum in coordinate $r$ is finite by step 1.1 and the product bounds $\sum_{n} \prod_{s} h_{s} (n_{s}) = \prod_{s} (1 + a_{s}) < \infty$ . Apply [F4] to finite partial sums on each coordinate interval; their values converge at one point by the absolute series and their derivative series converge uniformly by this bound. Thus the displayed series differentiates termwise. Subtracting the $n = 0$ term gives $∣ \partial_{t_{r}} b_{k} - \partial_{t_{r}} a_{k} ∣ \leq 2 π \sum_{n \neq 0} ∣ k_{r} + n_{r} M_{r} ∣ \prod_{s} h_{s} (n_{s}) = 2 π J_{k, r} .$ The last equality follows by summing the $n_{r} = 0$ term separately: the sum including $n = 0$ is $(∣ k_{r} ∣ + v_{r}) \prod_{s \neq r} (1 + a_{s})$ , from which the base $∣ k_{r} ∣$ is removed.

3.1F2step 2.1algebra

The physical gradient is $A^{- T}$ times the three fractional derivative components. Hence [F2] and step 2.1 give $∥ \nabla_{x} b_{k} - \nabla_{x} a_{k} ∥_{2} \leq ∥ A^{- T} ∥_{2} {(\sum_{r} (2 π J_{k, r})^{2})}^{1 / 2} = B_{k} .$ This proves the $P \geq 3$ branch.

4.1F2step 1.2algebra∎

Applying $A^{- T}$ to the three component bounds as in step 3.1 gives $∥ \nabla_{x} b_{k} - \nabla_{x} a_{k} ∥_{2} \leq ∥ A^{- T} ∥_{2} (\sum_{r} C_{k, r}^{2})^{1 / 2} = B_{k}$ . At a knot each one-sided limit satisfies the same coordinate bounds, and an average lies in their convex hull, so the declared branch value also satisfies them. No classical differentiability at a $P = 2$ knot is used.

Remarks

For $P = 2$ the absolute Fourier derivative majorant diverges at nonzero $u_{r}$ . The finite piecewise window derivative avoids that invalid rearrangement.
A tighter certified enclosure may subdivide the closed mesh boxes and bound the finite piecewise expression on each box. The explicit $C_{k, r}$ above is a valid initial enclosure and does not require a supremum oracle.

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The alpha-independent Ewald potential and renormalized energy

Statement

Assume Countable Choice and neutral finite point-source data $(A, V, Λ, κ_{k}, (x_{i}, q_{i})_{i = 1}^{n})$ from Neutral point sources and Fourier data on a three-dimensional torus, with pairwise distinct source positions in $T_{A}$ . For $α > 0$ put $σ_{k} = \sum_{i} q_{i} e^{- i κ_{k} \cdot x_{i}}, c_{k} (α) = \frac{4 π}{V ∣ κ_{k} ∣^{2}} e^{- ∣ κ_{k} ∣^{2} / (4 α^{2})} (k \neq 0) .$ For every off-source point $x \notin ⋃_{i} (x_{i} + Λ)$ , the absolutely convergent Ewald potential $U_{α} (x) = \sum_{i} q_{i} \sum_{λ \in Λ} \frac{erfc (α ∣ x - x_{i} + λ ∣)}{∣ x - x_{i} + λ ∣} + \frac{4 π}{V} \sum_{k \neq 0} \frac{e^{- ∣ κ_{k} ∣^{2} / (4 α^{2})}}{∣ κ_{k} ∣^{2}} σ_{k} e^{i κ_{k} \cdot x}$ is the off-source representative of the unique mean-zero periodic solution of $- Δ u = 4 π ρ$ . Hence it is independent of $α$ .

Define the regular part at $x_{i}$ by ${FP}_{x_{i}} U_{α} = \lim_{x \to x_{i}} (U_{α} (x) - \frac{q_{i}}{∣ x - x_{i} ∣}) .$ The limit exists, is independent of $α$ , and the renormalized energy is $E = \frac{1}{2} \sum_{i} q_{i} {FP}_{x_{i}} U_{α} = \frac{1}{2} \sum_{i} \sum_{(j, λ) \neq (i, 0)} q_{i} q_{j} \frac{erfc (α ∣ x_{i} - x_{j} + λ ∣)}{∣ x_{i} - x_{j} + λ ∣} + \frac{1}{2} \sum_{k \neq 0} c_{k} (α) ∣ σ_{k} ∣^{2} - \frac{α}{\sqrt{π}} \sum_{i} q_{i}^{2} .$ The prime-exclusion written as $(j, λ) \neq (i, 0)$ omits only each particle's own zero-image singularity.

Facts & Assumptions

Given: Countable Choice; an invertible cell matrix $A$ , its covolume $V$ , the lattice $Λ$ , and the angular reciprocal lattice $κ_{k}$ ; a finite neutral source configuration with pairwise distinct torus positions; and $α > 0$ .

[A1]

Countable Choice is used by Gaussian lattice Poisson summation and by the periodic distribution uniqueness interface (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

For $ϕ_{s} (r) = erfc (α r) / r$ and $g_{α} = α^{3} π^{- 3 / 2} e^{- α^{2} ∣ x ∣^{2}}$ , $- Δ ϕ_{s} = 4 π δ_{0} - 4 π g_{α}$ , and $ϕ_{s} (r) = r^{- 1} - 2 α / \sqrt{π} + O (r^{2})$ (Gaussian screened Coulomb kernel and its self limit).

[F2]

The normalized Gaussian periodizes as $\sum_{λ \in Λ} g_{α} (x + λ) = V^{- 1} \sum_{k} e^{- ∣ κ_{k} ∣^{2} / (4 α^{2})} e^{i κ_{k} \cdot x}$ with absolute convergence (Poisson periodization of a normalized Gaussian on a full-rank lattice).

[F3]

The neutral source has a unique mean-zero periodic distribution solution of $- Δ u = 4 π ρ$ , with coefficients $\hat{u} (k) = 4 π σ_{k} / (V ∣ κ_{k} ∣^{2})$ for $k \neq 0$ and zero mode fixed to zero (The mean-zero Fourier distribution solves the neutral periodic Poisson equation).

[F4]

The shell count and finite Gaussian controllers apply to both $Λ$ and the angular reciprocal lattice; $T_{p} (β, M)$ is finite and has the stated finite upper controller (Uniform lattice-shell counts and explicit Gaussian tail controllers).

[F5]

The real cell translates tile $R^{3}$ and have volume $V$ (Neutral point sources and Fourier data on a three-dimensional torus, Fundamental parallelotopes of a lattice tile Euclidean space with covolume volume).

[F6]

$\int_{- \infty}^{\infty} e^{- t^{2}} d t = \sqrt{π}$ (The Gaussian integral $\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt{π}$ ).

[F7]

Integration by parts applies on every finite interval to the smooth Gaussian integrand (If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ ).

[F8]

A nonnegative integral over a product or countable partition may be evaluated by Tonelli's theorem (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F9]

A compact ball contains a nonnegative smooth bump supported in any prescribed larger open ball (A smooth bump between concentric Euclidean balls).

[F10]

The mean of every nonconstant lattice character on the cell is zero: $V^{- 1} \int_{F} e^{i κ_{k} \cdot x} d x = 0$ for $k \neq 0$ (Orthogonality of the lattice characters over a fundamental domain).

Proof

technique · direct

1.1givenA1F2F4algebra

Write $Q = \sum_{i} ∣ q_{i} ∣$ . For $α > 0$ define the periodized Gaussian source $γ_{α} (x) = \sum_{i} q_{i} \sum_{λ \in Λ} g_{α} (x - x_{i} + λ) .$ By [F2] applied to each translate and then summed over the finite source set, $γ_{α} (x) = \frac{1}{V} \sum_{k \in Z^{3}} e^{- ∣ κ_{k} ∣^{2} / (4 α^{2})} σ_{k} e^{i κ_{k} \cdot x} .$ Neutrality gives $σ_{0} = 0$ , so the $k = 0$ term vanishes. Both series are absolutely convergent; [F4] supplies uniform Gaussian tails for the lattice sums and their derivatives.

1.2givenF1F4F5F6F7F8F10algebra

Define $S_{α} (x) = \sum_{i} q_{i} \sum_{λ \in Λ} ϕ_{s} (∣ x - x_{i} + λ ∣)$ and $L_{α} (x) = \frac{4 π}{V} \sum_{k \neq 0} \frac{e^{- ∣ κ_{k} ∣^{2} / (4 α^{2})}}{∣ κ_{k} ∣^{2}} σ_{k} e^{i κ_{k} \cdot x}$ . The finitely many nonzero modes with $∣ κ_{k} ∣ < 1$ cause no difficulty, and [F4] bounds the remaining Gaussian shells; with $∣ σ_{k} ∣ \leq Q$ , the reciprocal series is absolutely and uniformly convergent. Thus termwise integration over the finite-volume cell is valid, and [F10] gives $\frac{1}{V} \int_{F} L_{α} (x) d x = \frac{4 π}{V} \sum_{k \neq 0} \frac{e^{- ∣ κ_{k} ∣^{2} / (4 α^{2})}}{∣ κ_{k} ∣^{2}} σ_{k} (\frac{1}{V} \int_{F} e^{i κ_{k} \cdot x} d x) = 0$ . For the real part, [F4], [F5], and Tonelli [F8] give $\int_{F} \sum_{λ \in Λ} ϕ_{s} (∣ x - x_{i} + λ ∣) d x = \int_{R^{3}} \frac{erfc (α ∣ y ∣)}{∣ y ∣} d y = 4 π \int_{0}^{\infty} r erfc (α r) d r = \frac{π}{α^{2}} .$ For the last equality, set $s = α r$ and use the tail identity in [F1]. Tonelli [F8] gives $\int_{0}^{\infty} s erfc (s) d s = \frac{2}{\sqrt{π}} \int_{0}^{\infty} e^{- t^{2}} (\int_{0}^{t} s d s) d t = \frac{1}{\sqrt{π}} \int_{0}^{\infty} t^{2} e^{- t^{2}} d t .$ Integration by parts [F7] on $[0, R]$ gives $\int_{0}^{R} t^{2} e^{- t^{2}} d t = - R e^{- R^{2}} / 2 + (1 / 2) \int_{0}^{R} e^{- t^{2}} d t$ ; the boundary term tends to zero since $e^{R^{2}} \geq R^{4} / 2$ for $R > 0$ . Taking $R \to \infty$ and using [F6] yields $\int_{0}^{\infty} t^{2} e^{- t^{2}} d t = \sqrt{π} / 4$ , so $\int_{0}^{\infty} s erfc (s) d s = 1 / 4$ . Multiplying by $4 π α^{- 2}$ gives the displayed value. Hence $\int_{F} S_{α} = (π / α^{2}) \sum_{i} q_{i} = 0$ , and $S_{α} + L_{α}$ has mean zero.

2.1F1F4step 1.1step 1.2algebra

On compact sets avoiding all source images, the real sum and each differentiated tail converge uniformly by the $erfc$ bounds of [F1] and the shell controller [F4]. The reciprocal series and every derivative converge uniformly because each derivative adds only a fixed power of $∣ κ_{k} ∣$ to a Gaussian tail. Thus both are continuous off-source, periodic, and their Fourier operations below are legitimate.

3.1givenF1F2F4step 1.1step 2.1algebra

The distributional identity in [F1], summed over the locally finite source images, gives $- Δ S_{α} = 4 π ρ - 4 π γ_{α}$ . Absolute tail control from [F4] lets the distribution pairing pass through the image sum. Termwise differentiation of the uniformly convergent reciprocal series for $L_{α}$ gives $- Δ L_{α} = \frac{4 π}{V} \sum_{k \neq 0} e^{- ∣ κ_{k} ∣^{2} / (4 α^{2})} σ_{k} e^{i κ_{k} \cdot x} = 4 π γ_{α},$ where the last equality is step 1.1 and $σ_{0} = 0$ . Therefore $- Δ (S_{α} + L_{α}) = 4 π ρ$ .

4.1A1F3F9step 2.1step 3.1step 1.2

The sum $U_{α} = S_{α} + L_{α}$ is a mean-zero periodic distribution solution by steps 3.1 and 1.2. Uniqueness [F3] makes it equal to the same Fourier distribution $u$ for every $α > 0$ . Its absolutely convergent representative is continuous off-source by step 2.1. If two such representatives differed at a point, one real or imaginary component of their difference would keep a strict sign on a small ball; choose a nonnegative smooth bump supported there using [F9]. Their distribution pairings with that bump would differ, contradicting equality to $u$ . Thus the displayed potential is independent of $α$ .

5.1F1F4step 2.1step 4.1algebra∎

Near $x_{i}$ , the unique term with its own zero image is $q_{i} ϕ_{s} (∣ x - x_{i} ∣)$ . By [F1], subtracting $q_{i} / ∣ x - x_{i} ∣$ leaves the finite limit $- 2 α q_{i} / \sqrt{π}$ from that term; every other real term is continuous near $x_{i}$ because positions are distinct modulo $Λ$ , and the reciprocal term is smooth. This proves existence and alpha-independence of the regular part by step 4.1. Multiplying those regular parts by $q_{i} / 2$ gives the real-space sum with $(j, λ) \neq (i, 0)$ and the self contribution $- α \sum_{i} q_{i}^{2} / \sqrt{π}$ . The reciprocal contribution is $\frac{1}{2} \sum_{i} q_{i} L_{α} (x_{i}) = \frac{1}{2} \sum_{k \neq 0} \frac{4 π}{V ∣ κ_{k} ∣^{2}} e^{- ∣ κ_{k} ∣^{2} / (4 α^{2})} (\sum_{i} q_{i} e^{i κ_{k} \cdot x_{i}}) (\sum_{j} q_{j} e^{- i κ_{k} \cdot x_{j}}) = \frac{1}{2} \sum_{k \neq 0} c_{k} (α) ∣ σ_{k} ∣^{2} .$ All rearrangements are absolutely convergent by [F4]. This is exactly the stated renormalized energy.

Remarks

The cell shape enters through $V$ and $κ_{k} = 2 π A^{- T} k$ ; no cubic-cell assumption is made.
Neutrality removes the $k = 0$ obstruction. The theorem fixes the mean-zero convention and includes no surface term for another boundary convention.
The real-space potential is singular at source images. The reciprocal series is smooth; any uniform field or potential error for the point-source model must exclude collisions or state a regularized source norm.

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Deterministic reciprocal particle-mesh Ewald potential, energy, and force bounds

Statement

Assume Countable Choice and neutral finite source data on the cell $A Z^{3}$ from Neutral point sources and Fourier data on a three-dimensional torus. Fix $α > 0$ , integers $M_{r} \geq 1$ , an order $P \geq 2$ , and a finite symmetric set $K \subset Z^{3} ∖ {0}$ whose signed representatives are injective modulo $(M_{1}, M_{2}, M_{3})$ and satisfy $∣ k_{r} ∣ < M_{r} / 2$ . Require $D (k) \neq 0$ for every $k \in K$ , and let $c_{k} = 4 π e^{- ∣ κ_{k} ∣^{2} / (4 α^{2})} / (V ∣ κ_{k} ∣^{2})$ be the positive Ewald coefficient. Use the assignment factors $a_{k}, b_{k}, ρ_{k}, z_{k}$ of Periodic cardinal-spline assignment and unitary mesh transform, and put $Q = \sum_{j} ∣ q_{j} ∣$ .

For a query point $x$ write $t = A^{- 1} x$ modulo $Z^{3}$ and define the retained exact and mesh reciprocal potentials by $U_{K} (x) = \sum_{k \in K} c_{k} Re (ρ_{k} \overline{a_{k} (t)}), U_{M, K} (x) = \sum_{k \in K} c_{k} Re (z_{k} \overline{b_{k} (t)}) .$ Define $E_{K} = \frac{1}{2} \sum_{k \in K} c_{k} ∣ ρ_{k} ∣^{2}$ and $E_{M, K} = \frac{1}{2} \sum_{k \in K} c_{k} ∣ z_{k} ∣^{2}$ . For each mode let $A_{k}$ be the source-alias controller in Exact three-dimensional spline assignment aliases and FFT normalization, and let $B_{k}$ be the derivative-alias controller in Derivative-window alias bounds including the P=2 knot branch.

Let a complex-vector derivative/gather value $d_{k} (t)$ satisfy $∥ d_{k} (t) - d_{k} (t) ∥_{2} \leq H_{k}$ for a stated $H_{k} \geq 0$ , where $d_{k}$ is the declared derivative branch of Periodic cardinal-spline assignment and unitary mesh transform and $\nabla_{x} = A^{- T} \nabla_{t}$ . The uncorrected mesh and exact reciprocal forces are $F_{M, i}^{d} = - q_{i} \sum_{k \in K} c_{k} Re (\overline{z_{k}} d_{k} (t_{i})), F_{K, i} = - q_{i} \sum_{k \in K} c_{k} Re (\overline{ρ_{k}} \nabla_{x} a_{k} (t_{i})) .$ Then for every query point $x$ , $∣ U_{M, K} (x) - U_{K} (x) ∣ \leq Q \sum_{k \in K} c_{k} A_{k} (2 + A_{k}), ∣ E_{M, K} - E_{K} ∣ \leq \frac{Q^{2}}{2} \sum_{k \in K} c_{k} A_{k} (2 + A_{k}),$ and for every particle $i$ , $∥ F_{M, i}^{d} - F_{K, i} ∥_{2} \leq ∣ q_{i} ∣ Q \sum_{k \in K} c_{k} [∣ κ_{k} ∣ A_{k} + (1 + A_{k}) B_{k} + (1 + A_{k}) H_{k}] .$

For the declared energy-gradient gather $d_{k} = d_{k}$ (so $H_{k} = 0$ ), the mesh force is $- \nabla_{x_{i}} E_{M, K}$ wherever the classical derivative exists. This holds everywhere for $P \geq 3$ and off the assignment knots for $P = 2$ ; at a $P = 2$ knot the selected one-sided derivative or its average is a procedure value, with the same bound but without a differentiability claim.

The optional diagonal self correction is $C_{M} = \frac{1}{2} \sum_{i} q_{i}^{2} \sum_{k \in K} c_{k} (∣ b_{k} (t_{i}) ∣^{2} - 1), E_{M, K}^{c o r r} = E_{M, K} - C_{M} .$ For its finite cyclic implementation, set $h_{k} = c_{k} / ∣ D (k) ∣^{2}$ on $K$ , $γ_{d} = \sum_{k \in K} h_{k} e^{- 2 π i \sum_{r} k_{r} d_{r} / M_{r}}, S_{i} = \sum_{ℓ, m \in G} w_{ℓ} (t_{i}) w_{m} (t_{i}) γ_{ℓ - m} .$ Then $γ$ is real and even, $S_{i} = \sum_{k \in K} c_{k} ∣ b_{k} (t_{i}) ∣^{2}$ , and, on every differentiable branch (or as the same declared one-sided branch at a $P = 2$ knot), $\nabla_{x_{i}} (S_{i} / 2) = \sum_{ℓ, m \in G} w_{ℓ} (t_{i}) \nabla_{x} w_{m} (t_{i}) γ_{ℓ - m} .$ Consequently this local cyclic-kernel expression computes $C_{M}$ and its derivative exactly in real arithmetic. The finite tensor-transform gathers also have the exact real-arithmetic forms $U_{M, K} (x) = Re \sum_{ℓ \in G} w_{ℓ} (t_{x}) u_{ℓ}, u_{ℓ} = \sum_{k \in K} \frac{c_{k} z_{k}}{D (k)} e^{2 π i \sum_{r} k_{r} ℓ_{r} / M_{r}},$ $F_{M, i} = - q_{i} Re \sum_{ℓ \in G} λ_{ℓ} \nabla_{x} w_{ℓ} (t_{i}), λ_{ℓ} = \sum_{k \in K} \frac{c_{k} \overline{z_{k}}}{\overline{D (k)}} e^{- 2 π i \sum_{r} k_{r} ℓ_{r} / M_{r}} .$ These identities hold with the declared derivative branch, including the selected one-sided $P = 2$ knot value. It satisfies $∣ E_{M, K}^{c o r r} - E_{K} ∣ \leq \frac{Q^{2} - \sum_{i} q_{i}^{2}}{2} \sum_{k \in K} c_{k} A_{k} (2 + A_{k}) .$ In the energy-gradient branch its exactly differentiated force is $F_{M, i}^{c o r r} = - q_{i} \sum_{k \in K} c_{k} Re (\overline{z_{k} - q_{i} b_{k} (t_{i})} d_{k} (t_{i})),$ and $∥ F_{M, i}^{c o r r} - F_{K, i} ∥_{2} \leq ∣ q_{i} ∣ (Q - ∣ q_{i} ∣) \sum_{k \in K} c_{k} [∣ κ_{k} ∣ A_{k} + (1 + A_{k}) B_{k}] .$ The corrected force is the negative classical energy gradient in the same differentiable cases; at a $P = 2$ knot it is only the declared branch value. No pointwise error estimate here replaces an RMS estimate, and no claim is made that a different spectral differentiation rule is the gradient of the mesh energy.

Facts & Assumptions

Given: Countable Choice; a neutral finite source configuration, $A, V$ , $κ_{k}$ , and $a_{k} (t) = e^{- i κ_{k} \cdot x}$ ; an order- $P$ periodic mesh assignment; a finite symmetric retained mode set with $D (k) \neq 0$ ; and a derivative/gather residual $H_{k}$ .

[A1]

Countable Choice is assumed by the Gaussian Poisson and mean-zero periodic Ewald suppliers (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The cell coordinates obey $x = A t$ , $κ_{k} = 2 π A^{- T} k$ , the source coefficient is $ρ_{k} = \sum_{j} q_{j} e^{- i κ_{k} \cdot x_{j}}$ , and the charges are real (Neutral point sources and Fourier data on a three-dimensional torus).

[F2]

The positive coefficient $c_{k} = 4 π e^{- ∣ κ_{k} ∣^{2} / (4 α^{2})} / (V ∣ κ_{k} ∣^{2})$ and exact retained reciprocal energy are those in the alpha-independent Ewald formula (The alpha-independent Ewald potential and renormalized energy).

[F3]

For each retained mode, $∣ b_{k} - a_{k} ∣ \leq A_{k}$ , $∣ b_{k} ∣ \leq 1 + A_{k}$ , and $∣ z_{k} - ρ_{k} ∣ \leq Q A_{k}$ ; $A_{k}$ has finite p-series remainder controllers (Exact three-dimensional spline assignment aliases and FFT normalization).

[F4]

The declared derivative branch obeys $∥ d_{k} - \nabla_{x} a_{k} ∥_{2} \leq B_{k}$ , including the finite selected branch at $P = 2$ knots (Periodic cardinal-spline assignment and unitary mesh transform, Derivative-window alias bounds including the P=2 knot branch).

[F5]

$\nabla_{x} a_{k} = - i κ_{k} a_{k}$ ; the derivative follows from the sine/cosine derivatives and the chain rule (Neutral point sources and Fourier data on a three-dimensional torus, The derivatives of sine and cosine are cosine and minus sine, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ).

[F6]

The exact retained reciprocal potential is $\sum_{k \in K} c_{k} ρ_{k} e^{i κ_{k} \cdot x}$ , and the reciprocal energy is $\frac{1}{2} \sum_{k \in K} c_{k} ∣ ρ_{k} ∣^{2}$ (The alpha-independent Ewald potential and renormalized energy).

[F7]

The mesh factors are finite sums of the assignment weights, $\nabla_{x} = A^{- T} \nabla_{t}$ , and the declared branch is classical wherever the derivative exists; the chain rule differentiates these factors componentwise there (Periodic cardinal-spline assignment and unitary mesh transform, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ).

[F8]

The periodic assignment weights are real, $b_{k} (t) = (\sum_{ℓ \in G} w_{ℓ} (t) e^{- 2 π i \sum_{r} k_{r} ℓ_{r} / M_{r}}) / \overline{D (k)}$ , and $D (- k) = \overline{D (k)}$ (Periodic cardinal-spline assignment and unitary mesh transform).

Proof

technique · direct

1.1givenA1F1F2F3F6algebra

Under the Countable Choice assumption [A1], the Ewald supplier gives the reciprocal potential and energy formulas [F6]. Because the charges and assignment weights are real, $c_{- k} = c_{k}$ , $ρ_{- k} = \overline{ρ_{k}}$ , $a_{- k} = \overline{a_{k}}$ , $D (- k) = \overline{D (k)}$ , $b_{- k} = \overline{b_{k}}$ , and $z_{- k} = \overline{z_{k}}$ . Thus the sums over symmetric $K$ are real and their displayed real-part forms agree with the reciprocal potentials. By [F3], $∣ ρ_{k} ∣ \leq Q$ , $∣ z_{k} - ρ_{k} ∣ \leq Q A_{k}$ , and $∣ z_{k} ∣ \leq Q (1 + A_{k})$ . Since $e^{i κ_{k} \cdot x} = \overline{a_{k} (t)}$ , [F6] gives the exact reciprocal potential. For each mode, $∣ z_{k} \overline{b_{k}} - ρ_{k} \overline{a_{k}} ∣ \leq ∣ z_{k} - ρ_{k} ∣ ∣ a_{k} ∣ + ∣ z_{k} ∣ ∣ b_{k} - a_{k} ∣ \leq Q A_{k} + Q (1 + A_{k}) A_{k} = Q A_{k} (2 + A_{k})$ using $∣ a_{k} ∣ = 1$ . Sum with $c_{k} > 0$ to obtain the potential bound.

1.2F2F3algebra

The identity $∣ ∣ z_{k} ∣^{2} - ∣ ρ_{k} ∣^{2} ∣ \leq ∣ z_{k} - ρ_{k} ∣ (∣ z_{k} ∣ + ∣ ρ_{k} ∣)$ and [F3] give $∣ ∣ z_{k} ∣^{2} - ∣ ρ_{k} ∣^{2} ∣ \leq Q A_{k} (Q (1 + A_{k}) + Q) = Q^{2} A_{k} (2 + A_{k})$ . Multiplying by $c_{k} / 2$ and summing proves the energy bound.

1.3F3F4F5F6F7algebra

The mode sums are finite. For $P \geq 3$ , and for $P = 2$ off the assignment knots, [F4] supplies the classical derivatives. Since $z_{k} = \sum_{j} q_{j} b_{k} (t_{j})$ by [F3], [F7] gives $\nabla_{x_{i}} z_{k} = q_{i} \nabla_{x} b_{k} (t_{i})$ . Differentiating $\frac{1}{2} c_{k} ∣ z_{k} ∣^{2}$ yields $q_{i} c_{k} Re (\overline{z_{k}} \nabla_{x} b_{k} (t_{i}))$ , so negating and summing proves $F_{M, i}^{\nabla} = - \nabla_{x_{i}} E_{M, K}$ in those differentiable cases. Also $\nabla_{x_{i}} ρ_{k} = q_{i} \nabla_{x} a_{k} (t_{i})$ by [F5]; differentiating $\frac{1}{2} c_{k} ∣ ρ_{k} ∣^{2}$ gives $q_{i} c_{k} Re (\overline{ρ_{k}} \nabla_{x} a_{k} (t_{i}))$ , so the negative derivative is exactly $F_{K, i}$ . At a $P = 2$ knot the chosen derivative is a procedure value, not a classical derivative.

1.4givenF3F4F5algebra

Decompose the force integrand as $\overline{z_{k}} d_{k} - \overline{ρ_{k}} \nabla_{x} a_{k} = \overline{(z_{k} - ρ_{k})} \nabla_{x} a_{k} + \overline{z_{k}} (d_{k} - \nabla_{x} a_{k}) + \overline{z_{k}} (d_{k} - d_{k})$ . By [F3], [F4], the given residual bound $H_{k}$ , and $∥ \nabla_{x} a_{k} ∥_{2} = ∣ κ_{k} ∣$ from [F5], its norm is at most $Q [∣ κ_{k} ∣ A_{k} + (1 + A_{k}) B_{k} + (1 + A_{k}) H_{k}]$ . Multiply by $∣ q_{i} ∣ c_{k}$ and sum to prove the uncorrected force bound.

1.5F2F3algebra

Expanding $∣ z_{k} ∣^{2}$ with $z_{k} = \sum_{j} q_{j} b_{k} (t_{j})$ shows that subtracting $C_{M}$ replaces the diagonal terms $q_{i}^{2} ∣ b_{k} (t_{i}) ∣^{2}$ by $q_{i}^{2}$ . In $∣ ρ_{k} ∣^{2}$ , the diagonal terms are $q_{i}^{2} ∣ a_{k} (t_{i}) ∣^{2} = q_{i}^{2}$ . The remaining difference is a sum over $i \neq j$ of $q_{i} q_{j} (b_{k} (t_{i}) \overline{b_{k} (t_{j})} - a_{k} (t_{i}) \overline{a_{k} (t_{j})}) / 2$ . Each parenthesized difference is bounded by $A_{k} (1 + A_{k}) + A_{k} = A_{k} (2 + A_{k})$ using [F3]; and $\sum_{i \neq j} ∣ q_{i} q_{j} ∣ = Q^{2} - \sum_{i} q_{i}^{2}$ . Multiplication by $c_{k}$ and summation proves the corrected energy estimate.

1.6F2F8algebra

Because $K$ is symmetric and $h_{- k} = h_{k}$ , the finite sum defining $γ$ is real and satisfies $γ_{- d} = γ_{d}$ . Put $B_{k} (t) = \sum_{ℓ \in G} w_{ℓ} (t) e^{- 2 π i \sum_{r} k_{r} ℓ_{r} / M_{r}}$ . Expanding the finite sums gives $S_{i} = \sum_{k \in K} h_{k} ∣ B_{k} (t_{i}) ∣^{2} = \sum_{k \in K} c_{k} ∣ b_{k} (t_{i}) ∣^{2}$ by [F8]. Subtracting $\frac{1}{2} \sum_{i} q_{i}^{2} \sum_{k} c_{k}$ therefore gives exactly $C_{M}$ .

1.7F7F8algebra

By [F8], $\overline{b_{k} (t)} = \sum_{ℓ} w_{ℓ} (t) e^{2 π i \sum_{r} k_{r} ℓ_{r} / M_{r}} / D (k)$ . Also differentiating the finite sum for $b_{k}$ gives $d_{k} (t) = \sum_{ℓ} \nabla_{x} w_{ℓ} (t) e^{- 2 π i \sum_{r} k_{r} ℓ_{r} / M_{r}} / \overline{D (k)}$ on a classical branch and on the declared one-sided branch. Substituting these identities into the finite mode formulas for $U_{M, K}$ and $F_{M, i}$ and interchanging the finite mode and grid sums yields exactly the two tensor-transform gather forms in the Statement.

2.1F7step 1.6algebra

Differentiate the finite sum $S_{i}$ on any declared derivative branch. Symmetry $γ_{ℓ - m} = γ_{m - ℓ}$ makes the two product-rule sums equal, so $\nabla_{x_{i}} (S_{i} / 2) = \sum_{ℓ, m} w_{ℓ} (t_{i}) \nabla_{x} w_{m} (t_{i}) γ_{ℓ - m}$ . The identity is algebraic in the one-sided polynomial derivatives as well, so it holds for the declared $P = 2$ knot branch values.

3.1F3F4F5F7step 1.3step 1.4step 2.1algebra∎

Where the energy-gradient derivative exists, [F7] gives $\nabla_{x_{i}} C_{M} = q_{i}^{2} \sum_{k} c_{k} Re (\overline{b_{k} (t_{i})} \nabla_{x} b_{k} (t_{i}))$ . By step 2.1 this is also the finite cyclic-kernel expression displayed in the Statement. Subtracting this gradient from that of $E_{M}$ yields the displayed corrected force. At a $P = 2$ knot the same algebra defines the selected corrected branch, without claiming it is a gradient. Put $Q_{i} = Q - ∣ q_{i} ∣ = \sum_{j \neq i} ∣ q_{j} ∣$ , $z_{k}^{'} = z_{k} - q_{i} b_{k} (t_{i})$ , and $ρ_{k}^{'} = ρ_{k} - q_{i} a_{k} (t_{i})$ . Since $Re (\overline{a_{k}} \nabla_{x} a_{k}) = Re (- i κ_{k}) = 0$ , the exact force is unchanged when $ρ_{k}$ is replaced by $ρ_{k}^{'}$ . The finite sums and [F3] give $∣ z_{k}^{'} - ρ_{k}^{'} ∣ \leq Q_{i} A_{k}$ and $∣ z_{k}^{'} ∣ \leq Q_{i} (1 + A_{k})$ . Repeating the force decomposition in step 1.4 with these primed coefficients and $H_{k} = 0$ proves the corrected force bound.

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Explicit real and reciprocal Ewald potential, energy, and force tail bounds

Statement

Assume Countable Choice and the neutral finite configuration and full-rank cell of Neutral point sources and Fourier data on a three-dimensional torus. Fix $α > 0$ . Let $Q = \sum_{i} ∣ q_{i} ∣$ , choose real and reciprocal cutoffs $R, K \geq 1$ , and put $M_{R} = ⌊ R ⌋$ , $M_{K} = ⌊ K ⌋$ . Let $C_{R} = C_{A}$ be the shell constant for $Λ = A Z^{3}$ and $C_{K} = C_{2 π A^{- T}}$ the shell constant for the angular reciprocal lattice. For the real-space cutoff, retain all $(i, j, λ)$ with $∣ x_{i} - x_{j} + λ ∣ \leq R$ except $(i, i, 0)$ . For the reciprocal cutoff, retain all $k \neq 0$ with $∣ κ_{k} ∣ \leq K$ . Use the $T_{3} (β, M)$ controller of Uniform lattice-shell counts and explicit Gaussian tail controllers. Then the omitted tails in the exact Ewald potential, energy, and particle force satisfy $B_{R}^{U} = \frac{Q C_{R}}{\sqrt{π} α} T_{3} (α^{2}, M_{R}), B_{R}^{E} = \frac{Q^{2} C_{R}}{2 \sqrt{π} α} T_{3} (α^{2}, M_{R}),$ $B_{R}^{F, i} = \frac{∣ q_{i} ∣ Q C_{R}}{\sqrt{π}} (2 α + α^{- 1}) T_{3} (α^{2}, M_{R}),$ and $B_{K}^{U} = \frac{4 π Q C_{K}}{V} T_{3} ((4 α^{2})^{- 1}, M_{K}), B_{K}^{E} = \frac{2 π Q^{2} C_{K}}{V} T_{3} ((4 α^{2})^{- 1}, M_{K}),$ $B_{K}^{F, i} = \frac{4 π ∣ q_{i} ∣ Q C_{K}}{V} T_{3} ((4 α^{2})^{- 1}, M_{K}) .$ For a certified finite value, replace each $T_{3}$ by the corresponding finite upper controller $U_{3}$ . The real force is evaluated only away from collisions; no field bound is asserted at a point source.

The exact force is the negative gradient of the renormalized energy. With $r_{i j λ} = ∣ x_{i} - x_{j} + λ ∣$ and $a_{k} (t_{i}) = e^{- i κ_{k} \cdot x_{i}}$ , it is $F_{i} = - q_{i} \sum_{j = 1}^{n} \sum_{λ \in Λ}^{'} q_{j} ϕ_{s}^{'} (r_{i j λ}) \frac{x_{i} - x_{j} + λ}{r_{i j λ}} - q_{i} \sum_{k \neq 0} c_{k} Re (\overline{σ_{k}} \nabla_{x} a_{k} (t_{i})) .$ The prime excludes only $(j, λ) = (i, 0)$ ; the absolutely convergent reciprocal sum is real by pairing $k$ and $- k$ .

Facts & Assumptions

Given: Countable Choice, a neutral finite source configuration on $A Z^{3}$ , $α > 0$ , $R, K \geq 1$ , and the real/reciprocal Ewald formulas of The alpha-independent Ewald potential and renormalized energy.

[A1]

Countable Choice is part of the Ewald formulas' Gaussian Poisson and periodic distribution prerequisites (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

$ϕ_{s} (r) = erfc (α r) / r \leq e^{- α^{2} r^{2}} / (\sqrt{π} α r^{2})$ and $∣ ϕ_{s}^{'} (r) ∣ \leq (2 α / \sqrt{π}) e^{- α^{2} r^{2}} / r + e^{- α^{2} r^{2}} / (\sqrt{π} α r^{3})$ (Gaussian screened Coulomb kernel and its self limit).

[F2]

Every translate of either lattice has at most $C_{B} (m + 1)^{3}$ points in the shell $m \leq ∣ x ∣ < m + 1$ , and the resulting Gaussian-polynomial tail is bounded by $T_{3}$ or its finite controller $U_{3}$ (Uniform lattice-shell counts and explicit Gaussian tail controllers).

[F3]

The reciprocal coefficient is $c_{k} = 4 π e^{- ∣ κ_{k} ∣^{2} / (4 α^{2})} / (V ∣ κ_{k} ∣^{2})$ and $∣ σ_{k} ∣ \leq Q$ (The alpha-independent Ewald potential and renormalized energy).

[F4]

The off-source potential and renormalized energy have the absolutely convergent Ewald formulas displayed in The alpha-independent Ewald potential and renormalized energy.

[F5]

Continuously differentiable partial sums whose derivatives converge uniformly on a closed interval and whose values converge at one point have a differentiable limit with the termwise derivative (If continuously differentiable functions converge at one point and their derivatives converge uniformly on a closed interval, then the functions converge uniformly to a differentiable function whose derivative is the derivative limit).

[F6]

Away from collisions, kernel and reciprocal summands are differentiable in the source coordinates by the chain rule and the sine/cosine derivatives (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , The derivatives of sine and cosine are cosine and minus sine).

[F7]

Uniform limits of continuous real functions are continuous, and a real map with continuous partial derivatives on a neighborhood is totally differentiable (The uniform limit of continuous real-valued functions on a metric space is continuous, If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative).

Proof

technique · direct

1.1A1F1F2F4F5

Fix a collision-free configuration and choose a small closed product of coordinate balls around its source positions, using local torus lifts, that remains collision-free. Finitely many nearby nonself images then stay a positive distance from the singularity. For large real-space shell index $m$ , [F1] bounds the kernel by a constant times $e^{- α^{2} m^{2}}$ and its first position derivatives by $(2 α + α^{- 1}) e^{- α^{2} m^{2}} / \sqrt{π}$ . [F2] bounds the shell count uniformly over every translate; since the source-difference vectors stay in a bounded set, $∣ λ ∣ \to \infty$ forces the corresponding distance to infinity uniformly. Thus the real energy series and each first-derivative series converge uniformly on this product. Finite partial sums are continuously differentiable and their values converge at one point by [F4]. Under [A1], apply [F5] along each real coordinate line to justify termwise differentiation of the real energy.

1.2A1F2F3F4F5F6

For the reciprocal energy, [F3] gives $∣ σ_{k} ∣ \leq Q$ and Gaussian decay of $c_{k}$ , while [F6] gives $∣ \nabla_{x_{i}} σ_{k} ∣ = ∣ q_{i} ∣ ∣ κ_{k} ∣$ . The derivative of each summand $\frac{1}{2} c_{k} ∣ σ_{k} ∣^{2}$ is bounded by $c_{k} Q ∣ q_{i} ∣ ∣ κ_{k} ∣$ . All nonzero modes in shells $m \geq 1$ therefore have a summable position-uniform majorant $C (m + 1)^{3} e^{- m^{2} / (4 α^{2})}$ by [F2]; the remaining bounded ball contains finitely many modes because the reciprocal lattice is full rank. The energy summands themselves are bounded by $Q^{2} c_{k} / 2$ , whose shell sum also converges by [F2]–[F3]. Thus the reciprocal energy and all its first-derivative series converge uniformly on the same product. Finite partial sums have values converging at one point by [F4], so [F5] gives their termwise derivatives along every coordinate line.

1.3givenA1F1F2F4algebra

Under Countable Choice [A1], use the Ewald representation [F4]. For $r > R$ , the shell index $m$ is at least $M_{R} \geq 1$ . By [F1], each omitted real potential term is at most $e^{- α^{2} m^{2}} / (\sqrt{π} α m^{2})$ . The shell count [F2] and $m^{- 2} \leq 1$ bound the total image tail for one source by $C_{R} (\sqrt{π} α)^{- 1} T_{3} (α^{2}, M_{R})$ . Summing $∣ q_{i} ∣$ gives $B_{R}^{U}$ .

1.4givenA1F2F3algebra

For a reciprocal mode in the omitted shell $m \leq ∣ κ_{k} ∣ < m + 1$ , with $m \geq M_{K} \geq 1$ , [F3] gives the potential summand bound $(4 π Q / V) e^{- m^{2} / (4 α^{2})} / m^{2}$ . Multiply by [F2], use $m^{- 2} \leq 1$ , and sum to obtain $B_{K}^{U}$ . The energy summand is one half the coefficient times $∣ σ_{k} ∣^{2} \leq Q^{2}$ , so the same count yields $B_{K}^{E}$ .

2.1step 1.1step 1.2F7

Each coordinate derivative obtained in steps 1.1–1.2 is a locally uniform limit of continuous finite-sum derivatives, hence is continuous by [F7]. The open collision-free configuration set is a finite-dimensional Euclidean open set in local lifts. Applying the continuous-partials clause of [F7] shows the Ewald energy is totally differentiable there, with the termwise derivatives computed below as its gradient.

2.2step 1.1F1F4F6algebra

In a finite real-space partial sum truncated by the symmetric condition $∣ λ ∣ \leq L$ , differentiating terms with $x_{i}$ in the first position gives one half of $q_{i} \sum_{j, λ}^{'} q_{j} \nabla ϕ_{s} (∣ x_{i} - x_{j} + λ ∣)$ . Terms with $x_{i}$ in the second position give $- \frac{1}{2} q_{i} \sum_{j, λ}^{'} q_{j} \nabla ϕ_{s} (∣ x_{j} - x_{i} + λ ∣)$ . By [F6] and the chain rule, $\nabla ϕ_{s} (∣ y ∣) = ϕ_{s}^{'} (∣ y ∣) y / ∣ y ∣$ for $y \neq 0$ , so this radial vector gradient is odd. Reindexing the second sum by $λ \mapsto - λ$ preserves the finite truncation and the excluded self term, and makes it equal to the first half. Passing to the limit is justified by step 1.1, so the full real energy gradient is $q_{i} \sum_{j, λ}^{'} q_{j} ϕ_{s}^{'} (r_{i j λ}) (x_{i} - x_{j} + λ) / r_{i j λ}$ .

3.1step 1.2step 2.2F3F4F6algebra

For each reciprocal mode, [F6] gives $\nabla_{x_{i}} σ_{k} = - i q_{i} κ_{k} a_{k} (t_{i})$ . In real and imaginary coordinates, $\nabla ∣ σ_{k} ∣^{2} = 2 Re (\overline{σ_{k}} \nabla σ_{k})$ , so differentiating a finite reciprocal partial energy gives $q_{i} \sum_{k} c_{k} Re (\overline{σ_{k}} \nabla_{x} a_{k} (t_{i}))$ . Step 1.2 justifies passage to the full sum. The constant self term has zero derivative. Negating the real gradient in step 2.2 and this reciprocal gradient gives the force formula in the Statement; opposite reciprocal modes are conjugate pairs, so their sum is real.

3.2givenF1F2step 2.2algebra

The real energy has the factor $1 / 2$ and charge-pair absolute sum at most $Q^{2}$ , so the same shell estimate as in step 1.3 gives $B_{R}^{E}$ . The radial derivative bound [F1] and $r \geq m \geq 1$ imply $∣ ϕ_{s}^{'} (r) ∣ \leq (2 α + α^{- 1}) e^{- α^{2} m^{2}} / \sqrt{π}$ . The real force formula proved in step 2.2, the shell count [F2], and $∣ q_{i} ∣ \sum_{j} ∣ q_{j} ∣ = ∣ q_{i} ∣ Q$ give exactly $B_{R}^{F, i}$ .

4.1A1step 3.1F2F3F6givenalgebra

By the reciprocal force formula in step 3.1, each omitted reciprocal force mode has magnitude at most $(4 π ∣ q_{i} ∣ Q / V) e^{- ∣ κ_{k} ∣^{2} / (4 α^{2})} / ∣ κ_{k} ∣$ , using $∣ \nabla_{x} a_{k} ∣ = ∣ κ_{k} ∣$ from [F6]. Since $∣ κ_{k} ∣ \geq m \geq 1$ , this is at most $(4 π ∣ q_{i} ∣ Q / V) e^{- m^{2} / (4 α^{2})}$ . The shell count [F2] and sum over $m \geq M_{K}$ give $B_{K}^{F, i}$ .

5.1step 1.3step 3.2step 1.4step 4.1F2algebra∎

The shell bounds are uniform in every translate, so no source-position dependence was introduced. Replacing $T_{3}$ by its finite upper controller preserves each inequality. Adding the real and reciprocal omitted terms by the triangle inequality gives the total Ewald cutoff error for potential, energy, and force.

Remarks

The constants $C_{R}$ and $C_{K}$ depend on the cell shape through the two lattice bases. The estimate does not assume a cubic cell.
If the reciprocal retained set is not the full ball $∣ κ_{k} ∣ \leq K$ , every additional omitted mode, including omitted Nyquist modes, must be included in the reciprocal remainder.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Total deterministic Ewald and particle-mesh error certificate

Statement

Assume Countable Choice and a neutral finite point-source configuration on $A Z^{3}$ from Neutral point sources and Fourier data on a three-dimensional torus. Fix $α > 0$ , real and reciprocal cutoffs $R, K_{c} \geq 1$ , a mesh with integers $M_{r} \geq 1$ and order $P \geq 2$ , and a finite symmetric retained set $K \subseteq {k \in Z^{3} ∖ {0} : ∣ κ_{k} ∣ \leq K_{c}}$ that is injective modulo $(M_{1}, M_{2}, M_{3})$ and has $∣ k_{r} ∣ < M_{r} / 2$ and $D (k) \neq 0$ for every $k \in K$ . Put $Q = \sum_{i} ∣ q_{i} ∣$ and let $B_{K_{c}} = {k \in Z^{3} ∖ {0} : ∣ κ_{k} ∣ \leq K_{c}}, M_{K_{c}} = B_{K_{c}} ∖ K .$ This in-ball omitted set is finite. Write $M_{R} = ⌊ R ⌋$ , $M_{K} = ⌊ K_{c} ⌋$ , $C_{R} = C_{A}$ , and $C_{K} = C_{2 π A^{- T}}$ for the shell constants in Uniform lattice-shell counts and explicit Gaussian tail controllers. Choose integers $J_{R} \geq M_{R}$ , $J_{K} \geq M_{K}$ such that $2 α^{2} J_{R} (J_{R} + 1) \geq 3$ and $\frac{1}{2 α^{2}} J_{K} (J_{K} + 1) \geq 3$ . Let $U_{3} (β, M, J)$ be that supplier's finite upper controller for $\sum_{m = M}^{\infty} (m + 1)^{3} e^{- β m^{2}}$ , and put $\begin{aligned} {\overline{B}}_{R}^{U} & = \frac{Q C_{R}}{\sqrt{π} α} U_{3} (α^{2}, M_{R}, J_{R}), & {\overline{B}}_{R}^{E} & = \frac{Q^{2} C_{R}}{2 \sqrt{π} α} U_{3} (α^{2}, M_{R}, J_{R}), & {\overline{B}}_{R}^{F, i} & = \frac{∣ q_{i} ∣ Q C_{R}}{\sqrt{π}} (2 α + α^{- 1}) U_{3} (α^{2}, M_{R}, J_{R}), \\ {\overline{B}}_{K}^{U} & = \frac{4 π Q C_{K}}{V} U_{3} ((4 α^{2})^{- 1}, M_{K}, J_{K}), & {\overline{B}}_{K}^{E} & = \frac{2 π Q^{2} C_{K}}{V} U_{3} ((4 α^{2})^{- 1}, M_{K}, J_{K}), & {\overline{B}}_{K}^{F, i} & = \frac{4 π ∣ q_{i} ∣ Q C_{K}}{V} U_{3} ((4 α^{2})^{- 1}, M_{K}, J_{K}) . \end{aligned}$

For $ϕ_{s} (r) = erfc (α r) / r$ , define the finite real-space potential at an off-source point by $S_{R} (x) = \sum_{i} q_{i} \sum_{\begin{array}{c} λ \in Λ \\ ∣ x - x_{i} + λ ∣ \leq R \end{array}} ϕ_{s} (∣ x - x_{i} + λ ∣),$ and the finite real energy by $E_{R} = \frac{1}{2} \sum_{i} \sum_{j} \sum_{\begin{array}{c} λ \in Λ \\ ∣ x_{i} - x_{j} + λ ∣ \leq R \end{array}}^{'} q_{i} q_{j} ϕ_{s} (∣ x_{i} - x_{j} + λ ∣),$ where the prime omits exactly $(j, λ) = (i, 0)$ . Let $F_{R, i} = - q_{i} \sum_{\begin{array}{c} 1 \leq j \leq n, λ \in Λ \\ (j, λ) \neq (i, 0), ∣ x_{i} - x_{j} + λ ∣ \leq R \end{array}} q_{j} ϕ_{s}^{'} (∣ x_{i} - x_{j} + λ ∣) \frac{x_{i} - x_{j} + λ}{∣ x_{i} - x_{j} + λ ∣} .$ The cutoff index set is held fixed in this force sum; the cutoff indicator is not differentiated. Set $\tilde{U} (x) = S_{R} (x) + U_{M, K} (x),$ ${\tilde{E}}^{χ} = E_{R} + E_{M, K}^{χ} - \frac{α}{\sqrt{π}} \sum_{i} q_{i}^{2}, {\tilde{F}}_{i}^{χ} = F_{R, i} + F_{M, i}^{χ},$ where $χ = u n c o r r$ uses the uncorrected energy and force of Deterministic reciprocal particle-mesh Ewald potential, energy, and force bounds, and $χ = c o r r$ uses its optional diagonal self-corrected branch. For the uncorrected branch suppose the derivative/gather residuals are $H_{k}$ ; the corrected branch uses the energy-gradient gather, so $H_{k} = 0$ .

Suppose the computed finite outputs $\hat{U} (x), {\hat{E}}^{χ}, {\hat{F}}_{i}^{χ}$ have certified arithmetic radii $ε_{U}, ε_{E}, ε_{F, i} \geq 0$ relative to $\tilde{U}, {\tilde{E}}^{χ}, {\tilde{F}}_{i}^{χ}$ , respectively. Define $\begin{aligned} η_{U} = & {\overline{B}}_{R}^{U} + {\overline{B}}_{K}^{U} + Q \sum_{m \in M_{K_{c}}} c_{m} + Q \sum_{k \in K} c_{k} A_{k} (2 + A_{k}) + ε_{U}, \\ η_{E}^{u n c o r r} = & {\overline{B}}_{R}^{E} + {\overline{B}}_{K}^{E} + \frac{Q^{2}}{2} \sum_{m \in M_{K_{c}}} c_{m} + \frac{Q^{2}}{2} \sum_{k \in K} c_{k} A_{k} (2 + A_{k}) + ε_{E}, \\ η_{E}^{c o r r} = & {\overline{B}}_{R}^{E} + {\overline{B}}_{K}^{E} + \frac{Q^{2}}{2} \sum_{m \in M_{K_{c}}} c_{m} + \frac{Q^{2} - \sum_{i} q_{i}^{2}}{2} \sum_{k \in K} c_{k} A_{k} (2 + A_{k}) + ε_{E}, \\ η_{F, i}^{u n c o r r} = & {\overline{B}}_{R}^{F, i} + {\overline{B}}_{K}^{F, i} + ∣ q_{i} ∣ Q \sum_{m \in M_{K_{c}}} c_{m} ∣ κ_{m} ∣ + ∣ q_{i} ∣ Q \sum_{k \in K} c_{k} [∣ κ_{k} ∣ A_{k} + (1 + A_{k}) B_{k} + (1 + A_{k}) H_{k}] + ε_{F, i}, \\ η_{F, i}^{c o r r} = & {\overline{B}}_{R}^{F, i} + {\overline{B}}_{K}^{F, i} + ∣ q_{i} ∣ Q \sum_{m \in M_{K_{c}}} c_{m} ∣ κ_{m} ∣ + ∣ q_{i} ∣ (Q - ∣ q_{i} ∣) \sum_{k \in K} c_{k} [∣ κ_{k} ∣ A_{k} + (1 + A_{k}) B_{k}] + ε_{F, i} . \end{aligned}$ Then, for every off-source $x$ and each particle $i$ , $∣ \hat{U} (x) - U_{α} (x) ∣ \leq η_{U}, ∣ {\hat{E}}^{χ} - E ∣ \leq η_{E}^{χ}, ∥ {\hat{F}}_{i}^{χ} - F_{i} ∥_{2} \leq η_{F, i}^{χ} .$ Here $U_{α}, E, F_{i}$ are the exact mean-zero periodic Ewald potential, renormalized energy, and particle force. The finite omitted-mode sums $M_{K_{c}}$ explicitly charge every mode in the radial ball that the mesh did not retain, including Nyquist modes. Missing arithmetic radii do not produce a certificate; no RMS estimate is substituted for these deterministic bounds.

For the formal exact-constant work count, let $N_{i m g}$ be the total number of candidate real-image indices inspected in the finite potential, energy, and force loops, and let $N_{c e r t}$ be the total number of mode and finite controller entries inspected in constructing the omitted-mode ledger and error certificate. Count each such outer-loop inspection as one work unit and count the transform arithmetic in the formal exact-constant model of Mixed-radix and Bluestein reductions compute every finite DFT in O(N log N) arithmetic. For fixed $P$ , the finite evaluation uses $O ⁣ (N_{g} \log (2 N_{g}) + n P^{3} + N_{i m g} + N_{c e r t})$ work units, with at most three tensor transforms before optional mesh-self correction. The correction adds $O ⁣ (N_{g} \log (2 N_{g}) + n P^{6})$ when its cyclic kernel is precomputed. These counts do not include interval precision, elementary-function evaluation, memory traffic, bit complexity, or the transform subroutine's integer factor search and phase evaluation; no operation count is used as an accuracy bound.

Facts & Assumptions

Given: Countable Choice; a neutral finite source configuration on a full-rank cell; $α > 0$ , $R, K_{c} \geq 1$ , a finite admissible retained mode set $K$ ; integer mesh dimensions and $P \geq 2$ ; valid finite Gaussian tail-controller indices $J_{R}, J_{K}$ ; and actual arithmetic radii for the finite computations.

[A1]

Countable Choice is required by the Gaussian periodization and mean-zero periodic Poisson theorem (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

$Λ = A Z^{3}$ , $V = ∣ \det A ∣$ , $κ_{k} = 2 π A^{- T} k$ , the data are neutral, and source positions are distinct (Neutral point sources and Fourier data on a three-dimensional torus).

[F2]

The exact off-source potential and renormalized energy are the absolutely convergent Ewald formulas with reciprocal coefficient $c_{k}$ and self term $- α \sum_{i} q_{i}^{2} / \sqrt{π}$ ; the exact particle force differentiates the real and reciprocal Ewald terms under the collision-free convention (The alpha-independent Ewald potential and renormalized energy, Explicit real and reciprocal Ewald potential, energy, and force tail bounds).

[F3]

The finite real/reciprocal Ewald omissions are bounded by the displayed $B_{R}^{U}, B_{R}^{E}, B_{R}^{F, i}, B_{K}^{U}, B_{K}^{E}, B_{K}^{F, i}$ formulas, and replacing each $T_{3}$ by $U_{3}$ gives a finite certified upper bound (Explicit real and reciprocal Ewald potential, energy, and force tail bounds).

[F4]

The reciprocal particle-mesh potential, energy, and force aliases have the deterministic pointwise/per-particle bounds in Deterministic reciprocal particle-mesh Ewald potential, energy, and force bounds; the optional self-corrected energy/force branch has its corresponding improved retained-mode bounds there.

[F5]

A full-rank lattice has only finitely many points in a bounded ball; equivalently $∣ κ_{k} ∣ \leq K_{c}$ implies $∣ k ∣ \leq ∥ A^{T} ∥_{2} K_{c} / (2 π)$ (Full-rank lattices, covolume, and the dual lattice).

[F6]

The radii $ε_{U}, ε_{E}, ε_{F, i}$ bound the actual finite arithmetic outputs' differences from the exact-real finite truncated expressions defined in the Statement (given arithmetic enclosure interface).

[F7]

Differentiating the finite source coefficient gives $\nabla_{x_{i}} ρ_{k} = q_{i} \nabla_{x} a_{k} (t_{i}) = - i q_{i} κ_{k} a_{k} (t_{i})$ by the chain rule and sine/cosine derivatives (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , The derivatives of sine and cosine are cosine and minus sine, Neutral point sources and Fourier data on a three-dimensional torus).

[F8]

In the formal exact-constant model, an arbitrary-length one-dimensional DFT is exact and costs $O (M \log (2 M))$ arithmetic operations at length $M$ ; integer factor search and interval/phase evaluation are excluded (Arbitrary-length mixed-radix and Bluestein DFT, Mixed-radix and Bluestein reductions compute every finite DFT in O(N log N) arithmetic).

Proof

technique · direct

1.1F1F5algebra

From [F1], $κ_{k} = 2 π A^{- T} k$ , so $∣ κ_{k} ∣ \leq K_{c}$ implies $∣ k ∣ \leq ∥ A^{T} ∥_{2} K_{c} / (2 π)$ . Only finitely many integer vectors lie in that bounded ball by [F5]. Therefore $B_{K_{c}}$ and $M_{K_{c}}$ are finite, and all missing in-ball sums in the Statement are finite.

1.2givenF3

Choose $J_{R}, J_{K}$ as in the Statement. The finite Gaussian controllers $U_{3}$ bound the exact real and reciprocal shell tails by [F3], giving ${\overline{B}}_{R}^{U}, {\overline{B}}_{R}^{E}, {\overline{B}}_{R}^{F, i}, {\overline{B}}_{K}^{U}, {\overline{B}}_{K}^{E}, {\overline{B}}_{K}^{F, i}$ .

1.3A1F1F2F5F7algebra

For each omitted in-ball mode $m \in M_{K_{c}}$ , $∣ ρ_{m} ∣ \leq Q$ and $∣ e^{i κ_{m} \cdot x} ∣ = 1$ give potential magnitude at most $c_{m} Q$ . Its energy term is $\frac{1}{2} c_{m} ∣ ρ_{m} ∣^{2} \leq \frac{1}{2} c_{m} Q^{2}$ . By [F7], differentiating its finite reciprocal energy term gives exact force $- q_{i} c_{m} Re (\overline{ρ_{m}} \nabla_{x} a_{m} (t_{i}))$ , whose magnitude is at most $∣ q_{i} ∣ c_{m} ∣ ρ_{m} ∣ ∣ κ_{m} ∣ \leq ∣ q_{i} ∣ Q c_{m} ∣ κ_{m} ∣$ . Summing these inequalities over the finite omitted set accounts for every in-ball mode, including modes excluded by Nyquist restrictions.

1.4F4

The retained-mode mesh errors are bounded by [F4]. In the uncorrected energy branch this is $\frac{1}{2} Q^{2} \sum_{k \in K} c_{k} A_{k} (2 + A_{k})$ ; in the corrected branch it is $\frac{1}{2} (Q^{2} - \sum_{i} q_{i}^{2}) \sum_{k \in K} c_{k} A_{k} (2 + A_{k})$ . The force contributions are the corresponding $∣ q_{i} ∣ Q$ and $∣ q_{i} ∣ (Q - ∣ q_{i} ∣)$ sums in [F4], and the retained potential contribution is $Q \sum_{k \in K} c_{k} A_{k} (2 + A_{k})$ .

2.1A1F2F3F4step 1.2step 1.3step 1.4algebra

The exact Ewald series is the sum of the full real-space term, all retained reciprocal modes, all in-ball omitted reciprocal modes, and the reciprocal tail beyond $K_{c}$ , with the exact self constant in the energy. The finite approximations include the real terms through $R$ , the selected mesh approximation on $K$ , and that same self constant. Thus their exact-real differences split into the real cutoff tail, the reciprocal shell tail, the finite omitted in-ball terms, and the retained mesh alias/differentiation error. Step 1.2 bounds the real and reciprocal cutoff tails, step 1.3 accounts for every omitted in-ball mode, and step 1.4 bounds the retained mesh error. By [F3], [F4], and the triangle inequality, the absolute values or norms of these parts sum to the first terms displayed in each $η$ .

3.1F6step 2.1algebra

The arithmetic assumptions [F6] give $∣ \hat{U} - \tilde{U} ∣ \leq ε_{U}$ , $∣ {\hat{E}}^{χ} - {\tilde{E}}^{χ} ∣ \leq ε_{E}$ , and $∥ {\hat{F}}_{i}^{χ} - {\tilde{F}}_{i}^{χ} ∥_{2} \leq ε_{F, i}$ . Applying the triangle inequality to step 2.1 yields the three stated certificates. If an arithmetic radius is unavailable, the corresponding inequality has no supplied numerical right-hand side and no certificate is returned.

4.1F8algebra∎

For fixed $P \geq 2$ , the finite support enclosure gives at most $P + 2 \leq 2 P$ candidates per coordinate. Thus each particle's spread and potential/force gathers use $O (P^{3})$ support tuples; grid initialization costs $O (N_{g})$ and is absorbed by the transform count. A tensor transform along coordinate $r$ consists of $N_{g} / M_{r}$ one-dimensional transforms of length $M_{r}$ . By [F8] this costs $O (N_{g} \log (2 M_{r}))$ , and summing over the three coordinates gives $O (N_{g} \log (2 N_{g}))$ . The uncorrected computation uses one forward transform and at most two inverse/gather transforms. The finite real-image and mode/controller loops contribute $N_{i m g} + N_{c e r t}$ by definition, proving the stated total count. The optional correction adds one transform to form its cyclic kernel and at most $(P + 2)^{6} = O (P^{6})$ support-pair products per particle, which gives its stated cost. The arithmetic radii are certified separately and do not follow from this operation count.

AlgorithmAlgorithm: Literature-sourcedProof: Not applicableOpen item page →

Certified three-dimensional particle-mesh Ewald evaluator

Definition

Assume Countable Choice. Inputs are a full-rank real cell matrix $A$ , a finite neutral configuration of pairwise-distinct torus positions $x_{i}$ and real charges $q_{i}$ , an Ewald parameter $α > 0$ , cutoffs $R, K_{c} \geq 1$ , integers $M_{r} \geq 1$ , an order $P \geq 2$ , a mesh-self-correction flag, an optional off-source query point $x$ , requested nonnegative tolerances for each requested output, and integer controller indices $J_{R} \geq ⌊ R ⌋$ , $J_{K} \geq ⌊ K_{c} ⌋$ , and $J_{A} \geq 1$ . The inputs must satisfy $2 α^{2} J_{R} (J_{R} + 1) \geq 3$ and $(2 α^{2})^{- 1} J_{K} (J_{K} + 1) \geq 3$ for the shell controllers. The arithmetic interface is one fixed instance of Operationwise real and complex floating point arithmetic with rounded square roots and fused multiply-add, with representable set $F$ and certified outward enclosures for every finite arithmetic node, elementary-function evaluation, transform phase, and final arithmetic radius. There is an optional positive integer enclosure-work cap $B_{e n c l}$ , defaulting to $1$ when omitted. Each non-transform validation, enclosure, or comparison request executes its supplied certificate lookup/evaluation program for at most $B_{e n c l}$ program instructions, including control-flow steps. Its numerical primitives are exact rational arithmetic/comparisons, fixed-format rounded operations, and reads of finite supplied certificate data. Elementary functions and refinement loops are implemented through these counted instructions, not as uncapped oracle calls. On exhaustion, unavailable data, a nonfinite bound, an uncertified required divisor or an unresolved comparison, return arithmetic-inconclusive, except where an explicit cutoff, transform, or window-denominator branch below names its more specific status. Do not retry with larger budgets inside a run. A larger cap may be supplied for another run; no accepted enclosure loses its required soundness. Supplied finite certificate data and the cap are part of the arithmetic interface, so this rule does not impose a precision or accuracy restriction on successful outputs. Every particle and optional query point also has a certified representative interval in $[0, 1)^{3}$ after reduction modulo the cell. Transform calls use common controls: an exact rational phase tolerance $τ$ with $0 < τ \leq 1 / 4$ , integer exponent limits $e_{\min} \leq e_{\max}$ , integer precision $p_{0} \geq 2$ , finite integer cap $p_{\max} \geq p_{0}$ , integer elementary-series cap $J_{p h a s e} \geq 1$ , and integer radix-two depth $d_{\max} \geq 0$ required by Arbitrary-length mixed-radix and Bluestein DFT. Every vector passed to a transform has real and imaginary coordinates in $F$ , and the outward enclosures include phase and arithmetic errors from the entire transform call. If $\sum_{i} q_{i} \neq 0$ is certified, return nonneutral. Otherwise check neutrality, distinctness, full rank, valid scalar/integer controls, and the required input enclosures using the bounded interface above. A certified violation of these input conditions returns invalid-input; unavailable or unresolved evidence returns arithmetic-inconclusive. If a query point is certified to be a source image, return collision for that potential query; if its off-source status is unresolved, return arithmetic-inconclusive. A controller inequality that is certified false is invalid-input; an unresolved one is arithmetic-inconclusive.

Compute a certified outward upper bound $B_{K}$ for $∥ A^{T} ∥_{2} K_{c} / (2 π)$ and set $N_{K} = ⌈ B_{K} ⌉$ . Enumerate every integer triple in the finite box $[- N_{K}, N_{K}]^{3}$ . For each nonzero $k$ , compute a certified interval $[d_{k}^{-}, d_{k}^{+}]$ for $∣ 2 π A^{- T} k ∣$ and an interval $[K_{c}^{-}, K_{c}^{+}]$ for $K_{c}$ . Include $k$ in $B_{K_{c}}$ if $d_{k}^{+} \leq K_{c}^{-}$ , exclude it if $d_{k}^{-} > K_{c}^{+}$ , and return cutoff-inconclusive if neither comparison is certified. Retain from $B_{K_{c}}$ exactly the signed modes satisfying $∣ k_{r} ∣ < M_{r} / 2$ for every coordinate and store them in $K$ . This set is finite, symmetric and injective modulo the mesh. Every other mode in $B_{K_{c}}$ is kept in the finite omitted-mode ledger $M_{K_{c}} = B_{K_{c}} ∖ K$ ; it is never assigned zero error.
Initialize the full grid $g$ to zero. Use each particle's supplied certified representative interval $t_{i} \in [0, 1)^{3}$ . For each coordinate $r$ , enclose $a = M_{r} t_{i, r}$ by $[a^{-}, a^{+}]$ ; if $a^{+} - a^{-} > 1$ , return arithmetic-inconclusive. Enumerate every integer $L_{r}$ from $⌈ a^{-} - P ⌉$ through $⌊ a^{+} ⌋$ . This finite list contains every translate with $0 \leq a - L_{r} \leq P$ and has at most $P + 2$ entries. For every triple $(L_{1}, L_{2}, L_{3})$ , evaluate $q_{i} \prod_{r} W_{P} (M_{r} t_{i, r} - L_{r})$ , reduce $L_{r}$ modulo $M_{r}$ to the grid residue $ℓ_{r}$ , add the term to $g_{ℓ}$ , and store the particle's finite support tuples for later gathers. This Cartesian list computes the periodic assignment exactly because each integer $L_{r}$ is uniquely $ℓ_{r} + j M_{r}$ . Accumulate all terms that wrap to the same residue; when $P > M_{r}$ , do not drop duplicate contributions. The support is the causal predecessor support of the window, not an unspecified symmetric nearest-node rule.
For each coordinate fiber, call Arbitrary-length mixed-radix and Bluestein DFT with the supplied common transform controls on its representable center vector. If an input cannot be represented, a call returns any inconclusive or range-failure status, or a normalization scale fails, return transform-inconclusive and retain that substatus. Apply the negative-sign unitary tensor DFT ${\hat{g}}_{M} (k) = N_{g}^{- 1 / 2} \sum_{ℓ \in G} g_{ℓ} e^{- 2 π i \sum_{r} k_{r} ℓ_{r} / M_{r}}, N_{g} = M_{1} M_{2} M_{3},$ by dividing each returned unnormalised coordinate transform by $\sqrt{M_{r}}$ , with outward enclosures for every scale operation. The required mathematical output convention is the unitary transform. The correctness-and-cost supplier proves the formal exact-constant transform identity; it is not used as a floating-point error bound.
For each retained $k$ , form the window multiplier $D (k) = \prod_{r = 1}^{3} e^{- π i P k_{r} / M_{r}} sinc (π k_{r} / M_{r})^{P}$ and obtain an outward lower bound for $∣ D (k) ∣$ . If any lower bound is nonpositive, return denominator-inconclusive without division. Otherwise set $z_{k} = \frac{\sqrt{N_{g}} {\hat{g}}_{M} (k)}{\overline{D (k)}}, c_{k} = \frac{4 π e^{- ∣ κ_{k} ∣^{2} / (4 α^{2})}}{V ∣ κ_{k} ∣^{2}} .$ Use the conjugated denominator forced by the negative-sign DFT and the causal window. The signed mode representative is used in every window factor; residues are used only as storage indices.
For every tensor-transform call in this step, use Arbitrary-length mixed-radix and Bluestein DFT with the same model and controls as in Step 3, divide each coordinate output by $\sqrt{M_{r}}$ , and propagate a failed or unrepresentable subcall or failed normalization as transform-inconclusive. Form mode arrays on the full mesh-frequency grid, placing retained signed modes in their unique residues and setting all other entries to zero. Set $v_{k}^{U} = c_{k} z_{k} / D (k)$ on $K$ . Apply the per-coordinate unitary normalization to the negative-sign tensor DFT of $\overline{v^{U}}$ , conjugate the output, and multiply by $\sqrt{N_{g}}$ ; the inverse-transform target array is $u_{ℓ} = \sum_{k \in K} v_{k}^{U} e^{2 π i \sum_{r} k_{r} ℓ_{r} / M_{r}}$ . If a query point was supplied, use its supplied representative $t_{x}$ and the finite-support routine of Step 2 to compute the off-source mesh potential $Re \sum_{ℓ \in G} w_{ℓ} (t_{x}) u_{ℓ}$ . For forces set $v_{k}^{F} = c_{k} \overline{z_{k}} / \overline{D (k)}$ on $K$ , zero elsewhere, and apply the same per-coordinate normalization to its negative-sign tensor DFT, then multiply by $\sqrt{N_{g}}$ to form $λ_{ℓ} = \sum_{k \in K} v_{k}^{F} e^{- 2 π i \sum_{r} k_{r} ℓ_{r} / M_{r}}$ . Evaluate $\nabla_{x} w_{ℓ}$ by differentiating the finite support products on the selected mesh box and applying $A^{- T}$ ; aggregate all tuple derivative contributions by residue, retaining duplicate wraps. Then $F_{M, i} = - q_{i} Re \sum_{ℓ \in G} λ_{ℓ} \nabla_{x} w_{ℓ} (t_{i})$ , gathering only the at most $(P + 2)^{3}$ candidate support contributions per particle. For $P = 2$ at a knot, use the one-sided derivative from the adjacent box immediately to the right in each knot coordinate (box index $⌊ M_{r} t_{r} ⌋$ modulo $M_{r}$ ), consistently for the full derivative vector and both correction factors. Determine this right-hand mesh-box index by the bounded certificate interface, allowing a supplied exact knot equality; if its floor/branch cannot be certified, return arithmetic-inconclusive. This is the selected branch from the assignment definition and is not asserted to be a classical gradient. Computed enclosures retain outward bounds on residual imaginary parts. The mesh energy is $E_{M, K} = \frac{1}{2} \sum_{k \in K} c_{k} ∣ z_{k} ∣^{2}$ .

If mesh-self correction is enabled, place $h_{k} = c_{k} / ∣ D (k) ∣^{2}$ on retained residues and zero elsewhere. Apply the same normalized negative-sign tensor DFT to $h$ and multiply by $\sqrt{N_{g}}$ to form the cyclic kernel $γ_{d} = \sum_{k \in K} h_{k} e^{- 2 π i \sum_{r} k_{r} d_{r} / M_{r}}$ . For each particle evaluate $S_{i} = \sum_{ℓ, m} w_{ℓ} (t_{i}) w_{m} (t_{i}) γ_{ℓ - m}$ over all support-residue pairs, retaining duplicate-wrap contributions, and set $C_{M} = \frac{1}{2} \sum_{i} q_{i}^{2} (S_{i} - \sum_{k \in K} c_{k})$ . Replace the mesh energy by $E_{M, K} - C_{M}$ . Replace the uncorrected force by $F_{M, i} + q_{i}^{2} \sum_{ℓ, m} w_{ℓ} (t_{i}) \nabla_{x} w_{m} (t_{i}) γ_{ℓ - m}$ , using the same selected derivative branch. The equality of this finite cyclic-kernel computation with the correction and its branch derivative is proved in Deterministic reciprocal particle-mesh Ewald potential, energy, and force bounds. Outward enclosures include the kernel transform, local products, derivative-window factors, and subtraction.
For each query/particle pair and particle pair, use certified representatives $t_{a}, t_{b} \in [0, 1)^{3}$ . If $∣ A (t_{a} - t_{b} + ℓ) ∣ \leq R$ , then $∣ ℓ ∣ \leq ∥ A^{- 1} ∥_{2} R + \sqrt{3}$ . Compute an outward upper bound $B_{R}$ for the right-hand side, set $N_{R} = ⌈ B_{R} ⌉$ , and for each required query or particle pair enumerate every integer index in $[- N_{R}, N_{R}]^{3}$ . For each candidate compute an outward interval $[r^{-}, r^{+}]$ for its image distance and an interval $[R^{-}, R^{+}]$ for $R$ ; include the image if $r^{+} \leq R^{-}$ , exclude it if $r^{-} > R^{+}$ , and return cutoff-inconclusive otherwise. Use these finite lists to compute the real potential when requested, real energy, and real force, excluding exactly the own zero-image term $(j, ℓ) = (i, 0)$ from energy and force before any singular kernel division. For every other retained image certify a positive lower distance bound before dividing; otherwise return arithmetic-inconclusive. Keep the force convention as the finite sum of differentiated retained pair kernels; do not differentiate the hard-cutoff membership indicator. Add the ordinary Ewald energy constant $- α \sum_{i} q_{i}^{2} / \sqrt{π}$ to the total energy. The point-source potential is only returned off all source images.
Compute $M_{R} = ⌊ R ⌋$ and $M_{K} = ⌊ K_{c} ⌋$ ; if either floor is not determined by its certified enclosure, return cutoff-inconclusive. Construct finite $U_{3}$ real and reciprocal shell controllers with the supplied $J_{R}, J_{K}$ . For each retained mode, compute outward bounds for $A_{k}$ and $B_{k}$ by summing the source/derivative alias terms through $∣ n_{r} ∣ \leq J_{A}$ and adding the supplier's finite p-series remainders; for $P = 2$ , use its finite derivative-window controller. Use $H_{k} = 0$ for the declared analytic-window gather in both force branches; place all finite arithmetic errors in the supplied radii $ε_{F, i}$ . Include every omitted mode in $M_{K_{c}}$ and the mesh-self correction if enabled. Add the actual outward arithmetic radii for all spread, transform, solve, gather, summation, and correction nodes. The outward enclosures for finite controllers themselves use the bounded enclosure interface above; its name does not authorize an unbounded oracle or refinement search. If a required arithmetic radius or controller enclosure is unavailable, return arithmetic-inconclusive. Otherwise retain the computed intervals and total error radii for the final output. Use the same bounded interface for tolerance comparisons: return certified when every requested radius is certified to meet its tolerance, certified-over-tolerance when at least one is certified to exceed it, and arithmetic-inconclusive if neither decision is certified. The total-certificate theorem proves the error bounds and formal operation-count properties for this evaluation; operation counts do not certify accuracy.

LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passOpen item page →

The certified particle-mesh Ewald evaluator terminates with finite outputs or an explicit status

Statement

Assume Countable Choice and use the finite input data and arithmetic interface of Certified three-dimensional particle-mesh Ewald evaluator, with its finite positive $B_{e n c l}$ , finite indices $J_{R}, J_{K}, J_{A}$ , and transform controls $τ, e_{\min}, e_{\max}, p_{0}, p_{\max}, J_{p h a s e}, d_{\max}$ . Every execution terminates, either with a declared failure/inconclusive status (including the potential-query collision status), or with finite computed intervals and error radii labelled certified or certified-over-tolerance. Missing or unresolved evidence cannot force an unbounded refinement search and cannot yield an accuracy certificate.

Once Steps 1 and 6 have obtained their finite integer bounds $N_{K}, N_{R}$ , the reciprocal candidate list has at most $(2 N_{K} + 1)^{3}$ triples and each real-pair candidate list has at most $(2 N_{R} + 1)^{3}$ triples. Each particle or query has at most $(P + 2)^{3}$ assignment-support tuples. There are at most four tensor transforms, including mesh-self correction, each consisting of $\sum_{r = 1}^{3} N_{g} / M_{r}$ one-dimensional calls. The optional correction uses at most $n (P + 2)^{6}$ support-pair terms. These finite bounds do not require the requested tolerances to be achievable. On either valid final-output branch, the error radii retain the guarantees of Total deterministic Ewald and particle-mesh error certificate for the same potential, energy, force, and optional mesh-self correction.

Facts & Assumptions

Given: A finite source count $n \geq 0$ , a finite optional query, the input controls and bounded enclosure programs specified by the evaluator. Countable Choice is assumed for its Ewald error suppliers; the termination argument uses only finite enumerations and integer induction.

[F1]

Certified three-dimensional particle-mesh Ewald evaluator: every non-transform validation/enclosure/comparison request is capped at $B_{e n c l}$ counted program instructions, including control flow; failed or unresolved requests return their specified status. Its integer boxes, support loops, tensor calls, correction pairs, and fixed controller truncations are the displayed instructions.

[F2]

Certified dyadic enclosures for unit-circle phases tries only $p_{0}, \dots, p_{\max}$ and uses series through $J_{p h a s e}$ and two fixed elementary expression DAGs in each attempt. Certified range evaluation for elementary real and complex functions evaluates a finite DAG once in topological order, with fixed series degree, finite domain pieces and finite canonical trigonometric-cell lists, returning a status on an unresolved piece. Its square-root endpoints use Exact dyadic bisection for square-root endpoints with request $n = p$ , and its arithmetic and endpoint conversions follow Outward-rounded interval, rectangle and ball operations and Dyadic floating-point format and directed rounding.

[F3]

Computed unitary radix-two FFT with certified twiddle phases decrements its nonnegative depth cap on recursive calls and otherwise returns at the cap. The cyclic mode of Computed FFT cyclic convolution and zero-padded linear convolution makes three such FFT calls and finitely many pointwise operations. All rounded scalar arithmetic has the success or range-failure behavior of Operationwise real and complex floating point arithmetic with rounded square roots and fused multiply-add.

[F4]

Arbitrary-length mixed-radix and Bluestein DFT has base length one, power-of-two calls to [F3], composite calls at proper factor lengths, and a prime branch padded to the least power of two at least $2 N - 1$ , using only the cyclic convolution in [F3] and the phases in [F2].

[F5]

Causal cardinal spline window defines $W_{P}$ by a finite sum of $P + 1$ truncated-power terms. The assignment and derivative suppliers provide the corresponding finite polynomial/one-sided derivatives and finite alias controllers, including the separate $P = 2$ branch (Periodic cardinal-spline assignment and unitary mesh transform, Exact three-dimensional spline assignment aliases and FFT normalization, Derivative-window alias bounds including the P=2 knot branch).

[F6]

Uniform lattice-shell counts and explicit Gaussian tail controllers defines $U_{3} (β, M, J)$ by a sum from $M$ through $J$ and four integral-controller terms $I_{0}, \dots, I_{3}$ , with $I_{s}$ depending only on $I_{s - 2}$ . The finite alias remainders in [F5] are explicit formulas, not remaining infinite-series evaluations.

[F7]

Total deterministic Ewald and particle-mesh error certificate proves the potential, energy, and force radii, for both self-correction branches, conditional on the actual outward arithmetic enclosures. It is used here only after termination has been proved, not to infer termination from a conditional work count.

Proof

1.1givenF1F3algebra

Each non-transform request [F1] either returns its supplied certified data or consumes at most the positive finite $B_{e n c l}$ instruction budget and exits. Each counted instruction on finite rational data or fixed-format operands terminates, with the model's range-failure outcome when appropriate. Thus no request can perform unbounded precision refinement, wait for an undecided equality, or search until a valid radius appears. Initial validation uses finitely many particle pairs and scalar/integer conditions. It either exits with a declared status, or supplies the finite data needed by the first step.

1.2F2givenalgebra

A phase call has at most $p_{\max} - p_{0} + 1$ precision attempts by [F2]. At each attempt the Machin sums have $J_{p h a s e} + 1$ terms and rational tails. Its elementary DAGs are finite. Topological sorting removes a node at each successful iteration or reports a cycle, so it finishes. For every finite positive-endpoint logarithm interval, only finitely many dyadic bins lie between its endpoints; arctangent uses at most seven fixed pieces. For sine/cosine the algorithm computes finite $K = ⌈ 2 M_{X} / p_{-} ⌉ + 1$ when $p_{-} > 0$ and examines only $- K, \dots, K$ , otherwise returning a status. Every series has its fixed truncation degree; the exponential squaring count is the finite integer $\max (0, e_{\max} + 2)$ . Square-root endpoint bisection starts with integer gap $2^{p + 1}$ and at each iteration replaces it by at most its ceiling half, so at most $p + 1$ iterations suffice. Directed conversion of a rational endpoint searches a finite representable set using integer comparisons. All other rational endpoint arithmetic terminates, and unresolved domains or nonfinite conversions return statuses. Therefore each elementary call and each capped phase call terminates, including when its requested phase tolerance is unmet.

2.1F2F3step 1.2algebra

Induction on the integer $d_{\max}$ proves termination of each radix-two call: length one returns directly, depth zero at larger length returns inconclusive, and otherwise its two children have depth $d_{\max} - 1$ . Each finite butterfly list uses only scalar operations and the terminating phase calls of step 1.2. An inverse call adds only two finite conjugations around its forward call. Hence a radix-two call has a finite binary recursion tree, of at most $2^{d_{\max} + 1} - 1$ nodes. A cyclic convolution has just three such calls and a finite list of products/scalings, so it terminates too.

2.2F1step 1.1algebra

In Step 1 of the evaluator, obtaining the finite bound $B_{K}$ either succeeds in bounded work or exits by step 1.1. Its outward endpoint has an exactly computable integer ceiling $N_{K}$ . The box has $(2 N_{K} + 1)^{3}$ integer triples; each distance/cutoff decision uses a bounded request and either includes, excludes, or returns cutoff-inconclusive. Retained-mode restrictions are finite integer comparisons. Thus both the retained set and omitted-mode ledger are constructed in finite work. Their symmetry, strict mesh restrictions and injection modulo the mesh are preserved because successful comparisons certify the exact radial membership: two strictly interior signed representatives differing by a mesh multiple must coincide coordinatewise.

2.3F1F5step 1.1algebra

In Step 2, if an assignment-coordinate interval has width greater than one, the stated exit occurs. Otherwise its integer support endpoints are finite, and the interval from $a^{-} - P$ to $a^{+}$ has length at most $P + 1$ . Any interval of length at most $P + 1$ contains at most $P + 2$ integers, so the triple list has at most $(P + 2)^{3}$ entries. The finite formula for $W_{P}$ in [F5] uses integer powers, factorials, binomial coefficients, and $P + 1$ summands, all finite for the given integer $P$ . Each arithmetic/enclosure request terminates by step 1.1. Reduction modulo each positive $M_{r}$ and accumulation keep duplicate wraps rather than starting another loop. The same proof applies to the optional query support and to the finite polynomial derivatives. If a $P = 2$ selected right-hand box cannot be certified within its budget, the explicit arithmetic-inconclusive branch returns; an exact knot certificate selects the same one-sided derivative in both correction factors.

2.4F1step 1.1algebra

In Step 6 the bounded request for $B_{R}$ either exits or gives a finite ceiling $N_{R}$ . For each needed pair the box has $(2 N_{R} + 1)^{3}$ candidates. There are at most $n$ query/source pairs and $n^{2}$ particle pairs for each of the energy and force sums. For a true retained image, $∣ t_{a} - t_{b} ∣ \leq \sqrt{3}$ gives $∣ ℓ ∣ \leq ∥ A^{- 1} ∥_{2} R + \sqrt{3} \leq B_{R}$ , so the box misses none. Each candidate uses a bounded include/exclude/cutoff-inconclusive test. The own zero-image term is omitted before singular kernel evaluation; every other kernel division requires a certified positive distance bound or exits. Its elementary values and finite derivative arithmetic use step 1.1. The real potential, energy, force and ordinary Ewald self constant thus finish, without differentiating the cutoff indicator.

3.1F4step 1.2step 2.1algebra

For arbitrary transform length $N$ , trial division can examine $2, \dots, N - 1$ and thus terminates. A composite non-power-of-two branch has factors $1 < r, s < N$ , makes $r$ length- $s$ calls and $s$ length- $r$ calls, and a finite twiddle list. Strong induction on $N$ proves these calls terminate, with length one the base case. A power of two uses step 2.1. At a prime, start at one and double until the padding length is at least $2 N - 1$ . This search is finite since $2^{j} \geq j + 1$ by integer induction, and its least resulting power $M$ satisfies $M < 4 N$ . There are finitely many chirp phases, one cyclic convolution of length $M$ , and finitely many final products; steps 1.2 and 2.1 finish them. Thus all transform calls terminate with their actual supplied caps; no sufficient-depth or phase-success assumption is needed.

3.2F1F5F6step 1.1step 2.2algebra

Step 7's floor requests terminate or return cutoff-inconclusive. With determined floors, [F6] constructs the real controller using $J_{R} - M_{R} + 1$ summands and the reciprocal controller using $J_{K} - M_{K} + 1$ summands, plus four $I_{s}$ terms apiece. The $I_{s}$ recurrence reduces $s$ by two and stops at zero or one; no search for larger $J_{R}, J_{K}$ is performed. For each of finitely many retained modes, each one-dimensional source/derivative alias sum through $1 \leq ∣ j ∣ \leq J_{A}$ has $2 J_{A}$ terms, followed by the explicit p-series remainder. Their tensor products need no infinite summation; even a direct finite triple list would have at most $(2 J_{A} + 1)^{3}$ entries. For $P = 2$ , [F5] instead supplies the finite derivative-window bound. The omitted ledger is finite by step 2.2. All controller elementary values, arithmetic radii and final sums are bounded requests under step 1.1. This proves finite construction or an explicit arithmetic-inconclusive exit.

4.1F1F4F5step 1.1step 3.1step 2.2step 2.3algebra

The full grid has finite size $N_{g} = M_{1} M_{2} M_{3}$ . Each tensor transform has exactly $N_{g} / M_{r}$ coordinate fibers in direction $r$ ; every one-dimensional call terminates by step 3.1. Step 3 uses one tensor transform; Step 5 uses at most one potential transform, one force transform, and one optional kernel transform. Thus there are at most four tensor transforms, with finitely many normalizations. Failed calls/scales return transform-inconclusive; bounded denominator certification in Step 4 returns denominator-inconclusive before an unsafe division. Retained-mode and grid array construction are finite. All gathers visit the finite supports from step 2.3, and self correction visits at most $(P + 2)^{6}$ ordered tuple pairs per particle. These polynomial products, derivatives and finite sums therefore terminate while retaining the same energy, force, and diagonal correction formulas.

5.1F1F7step 1.1step 1.2step 2.1step 3.1step 2.2step 2.3step 4.1step 2.4step 3.2algebra∎

Finally there are finitely many requested output radii and tolerance comparisons. Each is bounded by [F1]; proven satisfaction gives certified, proven excess gives certified-over-tolerance, and an unresolved decision returns arithmetic-inconclusive. Steps 1.1–3.2 prove that every preceding branch or subcall finishes; their finite concatenation therefore terminates with the stated record. Divisor, branch and enclosure checks prevent undefined successful outputs. On a valid final-output branch the unchanged arithmetic-enclosure assumptions and finite formulas meet [F7], so its potential, energy, force and optional self-correction error guarantees apply. No work-count premise was used to infer termination.

5 · Examples, counterexamples and false statements

None yet.

Ewald periodic potentials and particle mesh error

1 · Prerequisites

2 · Summary

Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

Causal cardinal spline window

Definition

Neutral point sources and Fourier data on a three-dimensional torus

Definition

Remarks

Gaussian screened Coulomb kernel and its self limit

Statement

Facts & Assumptions

Proof

Remarks

Periodic cardinal-spline assignment and unitary mesh transform

Definition

Remarks

Poisson periodization of a normalized Gaussian on a full-rank lattice

Statement

Facts & Assumptions

Proof

Uniform lattice-shell counts and explicit Gaussian tail controllers

Statement

Facts & Assumptions

Proof

Remarks

Smooth torus test functions have rapidly convergent Fourier pairings

Statement

Facts & Assumptions

Proof

Exact three-dimensional spline assignment aliases and FFT normalization

Statement

Facts & Assumptions

Proof

Remarks

The mean-zero Fourier distribution solves the neutral periodic Poisson equation

Statement

Facts & Assumptions

Proof

Derivative-window alias bounds including the P=2 knot branch

Statement

Facts & Assumptions

Proof

Remarks

The alpha-independent Ewald potential and renormalized energy

Statement

Facts & Assumptions

Proof

Remarks

Deterministic reciprocal particle-mesh Ewald potential, energy, and force bounds

Statement

Facts & Assumptions

Proof

Explicit real and reciprocal Ewald potential, energy, and force tail bounds

Statement

Facts & Assumptions

Proof

Remarks

Total deterministic Ewald and particle-mesh error certificate

Statement

Facts & Assumptions

Proof

Certified three-dimensional particle-mesh Ewald evaluator

Definition

The certified particle-mesh Ewald evaluator terminates with finite outputs or an explicit status

Statement

Facts & Assumptions

Proof

5 · Examples, counterexamples and false statements

Sources