Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24
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.

FALSE: solids with equal parallel cross-sectional areas are congruent

Statement

False claim: if two bounded solids have equal areas in every pair of parallel horizontal sections, then the solids are congruent.

Facts & Assumptions

Given: The boxes E=[0,1]3 and F=[0,2]×[0,1/2]×[0,1].

[F1]

If two bounded Jordan sets have Jordan sections outside content-zero exceptional parameter sets and their ordinary sectional contents agree away from those sets, then the two sets have equal content (Cavalieri: Jordan content is the integral of sectional contents, and equal sections give equal content).

[F2]

An isometry is a bijection preserving every pairwise distance (Isometry, isometric embedding, and the subspace metric on a subset).

Refutation

technique · direct
1.1

At each height z∈[0,1], both boxes have a rectangular section of area 1; outside that range both sections are empty. Thus [F1] gives equal volume.

givenF1algebra
1.2

The diameters are 3 for E and 22+(1/2)2+12=21/2 for F. Since these are unequal and [F2] makes every isometry preserve diameter, the boxes are not congruent.

F2algebra
2.1

Hence equal parallel cross-sectional areas determine equal volume here but do not force congruence.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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