ExampleConstruction: Literature-sourcedVerification: Literature-sourcedPipeline-generatedprecheck passjudge 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.

A tail flip turns a generic real into its complement modulo finite

Statement

If a finite condition mentions coordinate $S_{n + 1}$ only at indices below $k_{0}$ , then flipping every bit $S_{n + 1} (k)$ for $k \geq k_{0}$ fixes the condition and changes $S_{n + 1}$ into its complement modulo the finite initial segment $k_{0} = {k : k < k_{0}}$ .

Facts & Assumptions

Given: A finite condition $p$ , natural numbers $n, k_{0}$ , and $(n + 1, k) \notin dom (p)$ whenever $k \geq k_{0}$ .

[F1]

The tail-complement automorphism fixes finitely supported names defines the relevant all-bit forcing automorphism and proves that it fixes the condition and all earlier-coordinate parameters.

Proof

technique · direct coordinate calculation

1.1

Let $A = {(n + 1, k) : k \geq k_{0}}$ and let $π_{A}$ toggle the value of a condition exactly on $A$ . By the Given hypothesis, $A \cap dom (p) = \emptyset$ , so no value of $p$ is changed and $π_{A} p = p$ . Coordinates other than $n + 1$ are fixed pointwise.

F1givenconstruct

2.1

Write $S = S_{n + 1}$ . For $k < k_{0}$ , the flip does not act and $k \in π_{A} S$ exactly when $k \in S$ . For $k \geq k_{0}$ , it toggles the generic bit and $k \in π_{A} S$ exactly when $k \notin S$ . Hence $π_{A} S △ (ω ∖ S) = {k : k < k_{0}} = k_{0} .$ When $k_{0} = 0$ this is exact complementation; when $k_{0} = 1$ the only possible discrepancy is bit $0$ . For every $k_{0}$ , the discrepancy is finite, while the flipped set is an infinite tail. No selection or Choice principle is used.

F1step 1.1∎

Depends on

The tail-complement automorphism fixes finitely supported names

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

Solomon Feferman, Some applications of the notions of forcing and generic sets, complete proof of Theorem 4.12, printed pp. 343–344 (standard reference, not scraped)