Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-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 an infinite field, a finite linear system has no solution, exactly one solution, or infinitely many solutions according to its pivots

Statement

Let FF be an infinite field. A finite system over FF has no solutions when its augmented column contains a pivot, exactly one solution when it is consistent and every variable column contains a pivot, and infinitely many solutions when it is consistent and has a nonpivot variable. Consistency is not implied by the pivot condition on the variable columns: over any field the system with matrix (10)\begin{pmatrix}1\\0\end{pmatrix} and right-hand side (01)\begin{pmatrix}0\\1\end{pmatrix} has a pivot in its single variable column and also a pivot in its augmented column, and has no solution.

Facts & Assumptions

Given: A finite system over an infinite field FF.

[L1]

RREF detects inconsistency and parametrises solutions by arbitrary values of the nonpivot variables (Reduced row echelon form detects consistency and parametrises every solution by the nonpivot variables).

[L2]

A set is finite when it is equinumerous with a natural number (The cardinality ∣A∣\lvert A\rvert of a finite set).

[L3]

A field supplies addition, multiplication and distinct 0,10,1, and each nonzero scalar is invertible (Field).

[L4]

An infinite set is one that is not finite; countability is a separate property and is not assumed here (Finite, countably infinite, countable, uncountable).

Proof

technique · direct
1.1

If the augmented column is a pivot column, [L1] gives no solution.

L1L2L3
2.1

If there is no augmented pivot and no free variable, [L1] determines every variable uniquely, so there is exactly one solution.

step 1.1L1
3.1

If a free variable exists, fix all other free variables and let that one range through FF. The parametrisation of [L1] assigns distinct solutions to distinct scalars, injecting the infinite set FF into the solution set; hence the solution set is not finite and is infinite.

step 2.1L1L4∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 36 results over 14 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