Alphabeta Math
RemarkRemark: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26
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.

What is and is not proved about rank⁡Wt,k(n) on this page

Remarks

This page proves the composition identity for the matrices Wt,k(n) and the rank formula for the point case t=1 in rank⁡QW1,k(n)=n for 1≤k≤n−1. It does not prove the general rank formula rank⁡QWt,k(n)=(nt) throughout the full stable range.

The reason is mathematical, not clerical. The natural induction on n through block decomposition loses one unit of the range at each step, and the boundary case is equivalent to the nonsingularity of the disjointness matrix. Standard proofs there use eigenvalues of the Johnson scheme or Gottlieb's construction, and neither belongs to this page's declared prerequisites. So the page records the point case and stops there.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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