ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck pendingverified 2026-10-23 (gpt-6.1-sol)

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.

The Bingham graph at zero contains nonzero tensors

Example

Let $M_{3} (R)$ denote the real $3 \times 3$ matrices and set $S_{0} = {D \in M_{3} (R) : D^{T} = D, tr D = 0}$ , the space of trace-free symmetric matrices with Frobenius product $D : E = tr (D^{T} E)$ and norm $∣ D ∣ = (D : D)^{1 / 2}$ . Let $τ_{y} > 0$ , let $D = 0 \in S_{0}$ , and use the Frobenius-norm Bingham graph whose zero-rate value is ${τ \in S_{0} : ∣ τ ∣ \leq τ_{y}}$ . The tensor

S = \frac{τ_{y}}{\sqrt{2}} (\begin{array}{ccc} 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 0 \end{array})

is nonzero, trace-free, and has Frobenius norm $τ_{y}$ , so $S \in B (0)$ . Thus zero strain rate does not imply zero deviatoric stress in this set-valued Bingham model.

Example references

the graph definition and zero-rate stress ball.

MIT OpenCourseWare, Lecture Notes on Fluid Dynamics, §1.6.2.

Facts & Assumptions

Given: A yield parameter $τ_{y} > 0$ , the tensor $D = 0$ , and the Bingham graph in the Example.

[F1]

Bingham yield graph and convex monotonicity: at $D = 0$ , $B (0) = {τ \in S_{0} : ∣ τ ∣ \leq τ_{y}}$ .

Verification

Given: The data in the Example and Facts & Assumptions.

1.1givenalgebra

The diagonal entries of $S$ sum to zero, so $S \in S_{0}$ ; its squared Frobenius norm is $τ_{y}^{2} / 2 + τ_{y}^{2} / 2 = τ_{y}^{2}$ , and $S \neq 0$ because $τ_{y} > 0$ .

2.1F1step 1.1

By [F1] and step 1.1, $∣ S ∣ = τ_{y}$ places $S$ in $B (0)$ .

3.1step 1.1step 2.1∎

The pair $D = 0, τ = S \neq 0$ is therefore an element of the graph, which refutes the implication that zero strain rate forces zero deviatoric stress for this model.

The Bingham graph at zero contains nonzero tensors

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources