Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07
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.

Gaps tending to infinity need not be Hadamard-lacunary

Statement refuted

“If an increasing integer sequence has gaps tending to infinity, then it is Hadamard-lacunary.”

Facts & Assumptions

Given: The increasing sequence λj=j2 for j≥1.

Counterexample

technique · direct gap and ratio calculation
1.1

Its successive gaps are [given, algebra] λj+1−λj=(j+1)2−j2=2j+1⟶∞.

givenalgebra
2.1

But [step 1.1, algebra] λj+1λj=(1+1j)2⟶1. Hence no fixed q>1 can satisfy the ratio condition eventually, much less at every index. This violates the definition in Hadamard-lacunary sequences and lacunary trigonometric series.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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