Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: Literature-sourcedprecheck passaudited 2026-08-11
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 coordinate shear preserves Jordan content by translating every section

Example

Fix i≠j and c∈R. The coordinate shear S(x1,…,xn)=(x1,…,xj+cxi,…,xn) preserves the Jordan content of every bounded Jordan set.

Facts & Assumptions

Given: The displayed shear S and bounded Jordan set E.

[L1]

For a bounded Jordan set whose sections are Jordan measurable outside a content-zero set of parameters, the completed sectional-content function is integrable and its integral is the content of the set (Cavalieri: Jordan content is the integral of sectional contents, and equal sections give equal content).

[L2]

Linear images scale Jordan content by the absolute determinant (A linear endomorphism of Rn sends bounded Jordan sets to bounded Jordan sets and scales their content by the absolute determinant).

Verification

technique · direct
1.1

The shear matrix is the identity with one off-diagonal entry c, so its determinant is 1. By [L2] the linear image S(E) is Jordan measurable and cont⁡(S(E))=∣1∣cont⁡(E)=cont⁡(E).

L2given
2.1

Sections show the same thing wherever [L1] applies. Hold all coordinates except xj fixed: the corresponding section of S(E) is the section of E translated by cxi, so its one-dimensional content is unchanged, and for a set whose sections are Jordan outside a content-zero parameter set [L1] integrates these equal values to the same total. This is a second reading of the result and not a second proof of it: a bounded Jordan set need not have Jordan sections outside a content-zero parameter set, so [L1] is not available for an arbitrary E and step 1.1 carries the statement.

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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