CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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.

Over $\mathbb F_2$ , the equation $x+y=0$ has exactly two solutions, so the infinite-field hypothesis is necessary

Statement refuted

The false extension is: over every field, a finite linear system has no solution, one solution, or infinitely many solutions. Over $\mathbb F_2$ , the equation $x+y=0$ has exactly two solutions.

Facts & Assumptions

Given: The equation $x+y=0$ with $x,y\in\mathbb F_2=\mathbb Z/2$ .

[L1]

The solution-count trichotomy assumes that the scalar field is infinite (Over an infinite field, a finite linear system has no solution, exactly one solution, or infinitely many solutions according to its pivots).

[L2]

A matrix equation records the same row equation as its linear system (Matrix equation $Ax=b$ , its solution set, consistency, homogeneous systems and the augmented matrix $[A\mid b]$ ).

[L3]

$\mathbb Z/2$ is a field (For every prime $p$ , the two operations on $\mathbb{Z}/p$ make it a field).

[L4]

$\mathbb Z/2$ consists of the congruence classes $0$ and $1$ (The congruence class $[a]_n$ and the quotient set $\mathbb{Z}/n$ ).

[L5]

Addition in $\mathbb Z/2$ is addition modulo $2$ (Addition and multiplication on $\mathbb{Z}/n$ by $[a]_n+[b]_n=[a+b]_n$ and $[a]_n[b]_n=[ab]_n$ ).

Counterexample

technique · direct

1.1

Exhausting $\mathbb F_2^2$ , the sums are $0+0=0$ , $0+1=1$ , $1+0=1$ , and $1+1=0$ ; hence precisely $(0,0)$ and $(1,1)$ solve the equation.

L2L3L4L5algebra

2.1

The solution set therefore has exactly two elements, so it is neither a singleton nor infinite. This refutes the extension and shows why [L1] requires an infinite field.

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 71 results over 22 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Hefferon, Linear Algebra, 4th ed., Ch. One, §III.2 (standard reference, not scraped)

Over F2\mathbb F_2F2​, the equation x+y=0x+y=0x+y=0 has exactly two solutions, so the infinite-field hypothesis is necessary