Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passaudited 2026-08-29
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: complexification doubles finite dimension

Statement

Complexification doubles finite dimension: for every finite-dimensional real vector space V,

dim⁡C(C⊗RV)=2dim⁡RV.

Facts & Assumptions

Given: A finite-dimensional real vector space V and its complexification VC.

[L1]

The complexification of Rn is canonically Cn through the standard inclusion (The standard embedding Rn↪Cn is the canonical complexification map).

[L2]

A real basis becomes a complex basis after complexification, so dim⁡C(C⊗RV)=dim⁡RV (A real basis becomes a complex basis after complexification, so dim⁡C(C⊗RV)=dim⁡RV).

Refutation

technique · direct
1.1

By [L2], dim⁡CVC=dim⁡RV for every finite-dimensional real V: the embedded image of a real basis is already a complex basis of the complexification.

L2
1.2

The concrete witness V=R confirms the correct value: by [L1] with n=1, VC≅C, so dim⁡CVC=1=dim⁡RR, not 2.

L1L2
2.1

The doubling behaviour belongs to realification, the reverse construction, which replaces complex scalars by real ones; complexification keeps the numerical dimension unchanged.

step 1.1step 1.2
3.1

Steps 1.1 and 1.2 contradict the claimed factor of 2, so the displayed statement is false.

step 1.1step 1.2∎

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