CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

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

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

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

Modified-energy decay alone does not imply physical-energy decay

Statement refuted

The scalar inequality ${\tilde{E}}^{1} \leq {\tilde{E}}^{0}$ , by itself, implies $E_{p h y s} (u^{1}) \leq E_{p h y s} (u^{0})$ .

Construction

Take a one-cell no-flux mesh with cell volume $∣ K ∣ = 1$ , so the gradient term is zero and the cell-volume mass is one, and set $ε = C_{0} = 1$ . Let $u^{0} = 1$ , $u^{1} = 0$ , $r^{0} = 1$ , and $r^{1} = 1 / 2$ . Then the modified scalar energy $r^{2}$ decreases from $1$ to $1 / 4$ , while the original double-well energy increases from $F_{1} (1) = 0$ to $F_{1} (0) = 1 / 4$ . These scalar pairs are a counterexample to the stated implication from an energy inequality alone; they are not asserted to be a trajectory of the SAV update equations.

Facts & Assumptions

Given: The physical double-well energy and the SAV modified energy definition.

[F1]

Allen–Cahn and Cahn–Hilliard double-well gradient flows: $F_{1} (s) = (s^{2} - 1)^{2} / 4$ is the original potential energy density.

[F2]

No-flux finite-volume and conforming realizations for phase fields: $E_{h} (u) = \frac{κ}{2} ⟨ A_{h} u, u ⟩_{h} + E_{c, h} (u) - E_{e, h} (u)$ ; on a one-cell no-flux mesh $A_{h} = 0$ and the potential term is $F_{1} (u)$ .

[F3]

The SAV step is uniquely solvable and dissipates a modified energy: the SAV theorem dissipates ${\tilde{E}}_{h} (u, r) = \frac{κ}{2} ⟨ A_{h} u, u ⟩_{h} + r^{2}$ .

Refutation

1.1F1F2givenalgebra

On one cell, [F2] gives $A_{h} = 0$ and $E_{p h y s} (u) = F_{1} (u)$ . Direct substitution using [F1] gives $F_{1} (1) = 0$ and $F_{1} (0) = 1 / 4$ .

2.1F3step 1.1algebra∎

The modified energies of the displayed pairs are ${\tilde{E}}^{0} = (r^{0})^{2} = 1$ and ${\tilde{E}}^{1} = (r^{1})^{2} = 1 / 4$ by [F3], so the modified energy strictly decreases while the physical energy strictly increases by step 1.1. Hence the proposed implication is false. The pairs are not claimed to satisfy the SAV update, so this refutes only the logical implication from modified-energy decay alone.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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

Sources

Jie Shen, Jie Xu, and Jiang Yang, The scalar auxiliary variable (SAV) approach for gradient flows, Journal of Computational Physics 353 (2018), 407–416 (standard reference, not scraped)