Alphabeta Math
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.
  • 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.

A floating component without a constant-detecting primal constraint remains singular

Example

For a floating square subdomain with the local matrix

K=12(2−10−1−12−100−12−1−10−12),

suppose no corner value (and no other functional that detects constants) is primal on that component. Its local energy matrix still has the constant vector in its kernel, so the local dual solve is singular. Omitting one corner while another corner remains primal does not produce this counterexample; the failure is the absence of any local anchor.

Facts & Assumptions

Given: The floating local square matrix Ki and a dual subproblem with no primal functional nonzero on constant vectors.

[F1]

Scalar finite-element substructuring model: a floating local matrix represents the sum of the local triangle diffusion energies; the displayed matrix is the four-node square instance.

[F2]

Floating local kernels and positive interior blocks: a connected floating local stiffness has exactly the constants as kernel.

[F3]

Primal constraints remove local floating kernels: a primal corner value removes the local constant mode; no such conclusion holds if the component has no anchor.

Verification

1.1givenF1algebra

Multiplying the displayed matrix from [F1] by 1 gives zero, so the nonzero vector 1 is in the local nullspace.

2.1givenstep 1.1F2algebra

Since the dual subproblem imposes no primal value or functional that changes on constants, 1 also satisfies every homogeneous local dual constraint. Thus its restricted quadratic form is singular, contradicting unique solvability if one attempted the local solve.

3.1givenF2F3step 1.1∎

If one corner value were retained as primal, a dual correction would have to vanish there; a constant vector with that value zero is the zero vector, so this particular null mode would be removed as in [F3]. The counterexample therefore concerns a whole floating component with no anchor, not omission of one among several anchors.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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