ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedjudge 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.

Rademacher-series threshold

Example

Assume countable choice and dependent choice, and let $(ϵ_{n})_{n \geq 1}$ be independent fair signs taking exactly the values $- 1$ and $1$ . For real $α$ , the series $\sum_{n \geq 1} ϵ_{n} n^{- α}$ converges almost surely exactly when $α > 1 / 2$ ; if $α \leq 1 / 2$ it diverges almost surely. Its absolute series converges exactly when $α > 1$ .

Facts & Assumptions

[F1]

Kolmogorov three-series theorem: Let $(X_{n})_{n \geq 1}$ be independent real random variables and fix $A > 0$ . Put $Y_{n} = X_{n} 1_{{∣ X_{n} ∣ \leq A}}$ . Then $\sum_{n} X_{n}$ converges almost surely if and only if all three conditions hold: $\sum_{n} P (∣ X_{n} ∣ > A) < \infty, \sum_{n} E Y_{n} converges in R, \sum_{n} Var (Y_{n}) < \infty .$ The conditions hold for some $A > 0$ if and only if they hold for every $A > 0$ . No moment assumption is imposed on the untruncated variables.

[F2]

Almost-sure convergence of an independent series is a zero-one event: Let $(X_{n})_{n \in N}$ be an independent sequence of real random variables. Then the event ${\sum_{n = 0}^{\infty} X_{n} converges}$ has probability $0$ or $1$ .

[F3]

Countably many independent copies of a prescribed law exist: Assume countable choice and dependent choice. Every probability measure $ν$ on $(S, Σ)$ is the common law of a countable independent family of $S$ -valued random elements.

[F4]

The p-series for a real exponent p converges exactly when p is greater than one: For every real $p$ , $\sum_{k \geq 1} \frac{1}{k^{p}} converges ⟺ p > 1.$

Verification

Given: The construction and assumptions above.

1.1

Under countable choice and dependent choice, construct IID copies of the probability law on the two-point space ${- 1, 1}$ , each point having mass $1 / 2$ . For $α > 0$ , the summands have magnitude $n^{- α} \leq 1$ . At cutoff $A = 1$ , the tail probabilities and truncated means are zero, and the truncated variances are $n^{- 2 α}$ .

F3givenalgebra

2.1

The three-series theorem and the real p-series test therefore give almost-sure convergence for $α > 1 / 2$ and rule out probability-one convergence for $0 < α \leq 1 / 2$ . In the latter range the convergence event is a tail event of an independent sequence, so its zero-one law forces its probability to be zero. In particular the boundary $α = 1 / 2$ has the divergent harmonic variance series.

F1F4F2step 1.1

3.1

If $α \leq 0$ , the magnitudes $n^{- α} \geq 1$ do not tend to zero at any point, so the partial sums cannot converge. Finally at every point the absolute series equals $\sum_{n} n^{- α}$ , which converges exactly for $α > 1$ by the p-series test. This also checks the absolute boundary $α = 1$ and the term-test boundary $α = 0$ .

F4step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

26 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

Example 2.5.7, p. 85 (standard reference, not scraped)