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

$X$ can be $c$ -sparse to $Y$ while $Y$ is not $c$ -sparse to $X$

Statement refuted

If $X$ is $c$ -sparse to $Y$ , then $Y$ is $c$ -sparse to $X$ .

Facts & Assumptions

Given: A real $c = \frac{1}{2}$ , a singleton $X = {x}$ , a set $Y = {y_{0}, y_{1}, y_{2}}$ , and the graph with the unique edge $x y_{0}$ .

[L1]

The directional definition says that $X$ is $c$ -sparse to $Y$ when every member of $X$ has at most $c ∣ Y ∣$ neighbours in $Y$ , and similarly with the roles reversed (Sparsity of one vertex set to another, and weak sparsity of a pair).

Counterexample

technique · constructive

1.1

The vertex $x$ has exactly one neighbour in $Y$ , and $1 \leq \frac{1}{2} ∣ Y ∣ = \frac{3}{2}$ , so [L1] makes $X$ $\frac{1}{2}$ -sparse to $Y$ .

L1givenconstruct

2.1

The vertex $y_{0}$ has one neighbour in $X$ , but $\frac{1}{2} ∣ X ∣ = \frac{1}{2}$ , so [L1] shows that $Y$ is not $\frac{1}{2}$ -sparse to $X$ .

step 1.1L1

3.1

Thus directional sparsity is not symmetric.

step 2.1discharge-construct∎

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.

XX can be cc-sparse to YY while YY is not cc-sparse to XX