ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

The common-height cone repair in the complement case

Example

In two binary trees, take cone roots of heights $1$ and $3$ . Extending the first root to the common height $3$ before restricting the dense frontiers produces a genuine common-height matrix.

Facts & Assumptions

Given: $T_{1} = T_{2} = 2^{< ω}$ , the roots $t_{1} = 0$ and $t_{2} = 101$ , and a finite $k \geq 0$ .

[F1]

Common-height cone extension and restriction preserve the adjusted density parameters. Finitistic trees, level products, density, and matrices

Verification

1.1

The roots have heights $1$ and $3$ , so they cannot themselves witness one $(h, k)$ -matrix. Put $h = 3$ , extend $t_{1}$ to $s_{1} = 011$ , and take $s_{2} = t_{2} = 101$ .

F1givenconstruct

2.1

Let $p = h + k$ and $B_{i} = T_{i} (p)$ . Then $B_{i}$ is $p$ -dense. Set $C_{1} = B_{1} \cap {u : 011 ≼ u}$ and $C_{2} = B_{2} \cap {u : 101 ≼ u}$ . Each $C_{i}$ is exactly the height- $(3 + k)$ frontier above $s_{i}$ , hence is $(3, k)$ -dense, and $C_{1} \times C_{2}$ is a $(3, k)$ -matrix.

F1step 1.1

3.1

For example, when $k = 2$ , $C_{1} = {01100, 01101, 01110, 01111}$ and $C_{2} = {10100, 10101, 10110, 10111}$ . Each listed set dominates all four height-5 nodes above its height-3 root; when $k = 0$ , the calculation instead gives the singleton sets ${011}$ and ${101}$ .

F1step 2.1

4.1

More generally, if $B_{i}$ is merely $(3 + k)$ -dense rather than the whole level, the same restrictions remain $(3, k)$ -dense: a height- $(3 + k)$ extension of $s_{i}$ is dominated by some member of $B_{i}$ , and that member automatically lies above $s_{i}$ . This is the exact common-height repair used in the complement case.

F1step 1.1step 2.1∎

Depends on

Finitistic trees, level products, density, and matrices

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Halpern–Läuchli, A partition theorem (1966), complement case in the proof of Theorem 1, p. 367; explicit binary-tree calculation (standard reference, not scraped)