LemmaStatement: AI-adaptedProof: AI-generated

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.

Elementary angle and logarithm estimates used in amplification

PDF p. 27: Proof of Theorem 1.10; PDF p. 30: Proof of Theorem 1.4

Statement

Let $\arcsin$ be the inverse of sine restricted to $[- π / 2, π / 2]$ , let $\log$ denote the natural logarithm, and let $⌈ x ⌉$ be the least integer greater than or equal to $x$ . Then $\arcsin x \leq π x / 2$ for $0 \leq x \leq 1$ , $⌈ x ⌉ \leq 2 x$ for $x \geq 1$ , $\log (1 + x) \geq 2 x / (2 + x)$ for $x \geq 0$ , and $\arcsin x \leq 1.05 x$ for $0 \leq x \leq 1 / 4$ .

Facts & Assumptions

[F1]

The inverse-sine and logarithm estimates in the Statement hold on their respective stated intervals (Angular order and elementary inverse-trigonometric bounds).

[F2]

The least integer $⌈ x ⌉$ at least a real $x$ exists (Geometric growth, truncation and total decrement budgets). For every real $x$ there is an integer $m$ with $m \leq x < m + 1$ (Integer part: for every real x there is exactly one integer m with m ≤ x < m + 1).

Proof

Given: The real variable $x$ lies in the interval specified for each inequality.

Goal: Prove the four stated inequalities.

1.1F1

The two inverse-sine inequalities and the logarithm inequality follow on their stated ranges directly from [F1].

2.1givenF2step 1.1algebra∎

If $x \geq 1$ , choose $m$ as in [F2]. Since $m + 1$ is an integer greater than $x$ , the least-integer property gives $⌈ x ⌉ \leq m + 1 \leq x + 1 \leq 2 x$ . Together with step 1.1, this proves all four inequalities.

The article uses these estimates in the proofs cited above. Its citation records attribution, while the proof here uses the stated suppliers.

Source attribution

The article uses these estimates in the proofs cited above. Its citation records attribution, while the proof here uses the stated suppliers.

Depends on

Used by · 1

Parameter and critical-step estimates in general amplification

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

Sources

Expander graphs are globally synchronizing — Abdalla, Bandeira, Kassabov, Souza, Strogatz, Townsend (2026) · PDF pp. 27 and 30, proofs of Theorems 1.10 and 1.4