ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

How statement and proof provenance work

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

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The energy-gradient mesh force matches the differentiated mesh energy

Example

Fix $α > 0$ and take the unit cubic cell, $q_{1} = 1$ at $x_{1} = (1 / 8, 1 / 8, 1 / 8)$ and $q_{2} = - 1$ at $x_{2} = (5 / 8, 1 / 8, 1 / 8)$ , a $4 \times 4 \times 4$ mesh, order $P = 3$ , and retained modes $K = {(1, 0, 0), (- 1, 0, 0)}$ . The sources are neutral and lie off mesh knots. The mesh force from the energy-gradient gather is the negative gradient of the finite mesh reciprocal energy; the optional self correction has the displayed differentiated force. The theorem's deterministic bound applies.

Facts & Assumptions

Given: Countable Choice, $α > 0$ ; the dipole configuration, $M_{1} = M_{2} = M_{3} = 4$ , $P = 3$ , the displayed signed modes, and the unitary mesh assignment.

[A1]

The Ewald reciprocal error theorem carries Countable Choice from the periodic Gaussian construction (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The DFT-deconvolved spread coefficient obeys $z_{k} = \sum_{i} q_{i} b_{k} (t_{i})$ with the stated normalization (Exact three-dimensional spline assignment aliases and FFT normalization).

[F2]

The mesh energy is $\frac{1}{2} \sum_{k \in K} c_{k} ∣ z_{k} ∣^{2}$ , and its declared force is the real-part gather; the corrected energy subtracts $C_{M}$ (Periodic cardinal-spline assignment and unitary mesh transform).

[F3]

For $P \geq 3$ the mesh force is the negative energy gradient; the self-corrected force and deterministic energy/force bounds are as stated in the reciprocal PME theorem (Deterministic reciprocal particle-mesh Ewald potential, energy, and force bounds).

Verification

technique · direct

1.1givenalgebra

The two positions are distinct and neutral. Each coordinate multiplied by $M_{r} = 4$ is either $1 / 2$ or $5 / 2$ , hence no coordinate is an integer and both particles are off assignment knots. The mode set is symmetric, injective modulo $(4, 4, 4)$ , nonzero, and satisfies $∣ k_{r} ∣ < 2$ . Since $sinc (π / 4) \neq 0$ , the required window denominators are nonzero.

1.2F1F2algebra

By [F1], $z_{k} = q_{1} b_{k} (t_{1}) + q_{2} b_{k} (t_{2})$ . With $P = 3$ , the factors $b_{k}$ are differentiable. The chain rule gives $\nabla_{x_{i}} z_{k} = q_{i} \nabla_{x} b_{k} (t_{i})$ . Therefore $\nabla_{x_{i}} (\frac{1}{2} c_{k} ∣ z_{k} ∣^{2}) = q_{i} c_{k} Re (\overline{z_{k}} \nabla_{x} b_{k} (t_{i}))$ . Summing over $k \in K$ and negating yields exactly the stated energy-gradient force.

2.1F2step 1.2algebra

Differentiating $C_{M} = \frac{1}{2} \sum_{j} q_{j}^{2} \sum_{k} c_{k} (∣ b_{k} (t_{j}) ∣^{2} - 1)$ with respect to $x_{i}$ gives $\nabla_{x_{i}} C_{M} = q_{i}^{2} \sum_{k} c_{k} Re (\overline{b_{k} (t_{i})} \nabla_{x} b_{k} (t_{i}))$ . Hence $- \nabla_{x_{i}} (E_{M} - C_{M}) = - q_{i} \sum_{k} c_{k} Re (\overline{z_{k} - q_{i} b_{k} (t_{i})} \nabla_{x} b_{k} (t_{i}))$ , the corrected force formula.

3.1A1F3step 2.1∎

Since $P = 3$ , this instance has classical derivatives at all positions. Applying [F3] to the same admissible data gives the deterministic retained-mode error certificate, including the corrected bound when the correction is selected. A spectral differentiation rule is a different force choice and does not inherit this energy-gradient identity; at $P = 2$ knots the assignment definition supplies a branch value but no classical-gradient claim.

Depends on

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Dependency tree · two levels

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