LemmaStatement: AI-generatedProof: 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.

Uniform squared-row control including dense graphs

Statement

In the setting of Uniform centered normalization across the full Bernoulli density range, fix $0 < δ \leq 1$ . On the event $∣ d (i) - μ ∣ < δ μ$ for all vertices, the squared row norm $r^{2} = \max_{i} \sum_{j} ∣ H_{i j} ∣^{2}$ is at most $1 + δ$ . If $g > 3 / δ^{2}$ , the probability of the complement of this event tends to zero uniformly over $g \log n \leq n p \leq n$ .

Facts & Assumptions

[F1]

The centered matrix and parameter range are Uniform centered normalization across the full Bernoulli density range.

[F2]

The exact-mean Chernoff union estimate, translated to center $n p$ , and its uniform logarithmic-density consequence are Chernoff bounds with the exact degree mean.

Proof

Given: The objects and hypotheses in the Statement.

Goal: Prove all stated conclusions.

1.1F1givenalgebra

For every row, direct expansion of its Bernoulli values gives $\sum_{j} ∣ H_{i j} ∣^{2} = [(1 - 2 p) d (i) + (n - 1) p^{2}] / μ$ . If $p \leq 1 / 2$ , its value on the good event is at most $(1 - 2 p) (1 + δ) + p \leq 1 + δ$ . If $p > 1 / 2$ , use the lower degree bound instead: its value is at most $(1 - 2 p) (1 - δ) + p = 1 - p + (2 p - 1) δ \leq 1 + δ$ . These calculations include $p = 1$ ; the row itself is then zero.

2.1F1F2step 1.1algebra∎

Write $p = γ \log n / n$ with $γ = n p / \log n \geq g$ . Supplier [F2] bounds the bad event by $2 n^{1 - γ a_{n}}$ , where $a_{n} = (n δ - 1)^{2} / (3 n (n - 1)) \to δ^{2} / 3$ . For fixed $g > 3 / δ^{2}$ , choose $b > 1$ with $g a_{n} \geq b$ eventually. The bound is then at most $2 n^{1 - b}$ independently of $p$ . Combine this with step 1.1.

Depends on

Used by · 1

Uniform constant-two adjacency control above a large logarithmic density

Dependency tree · 2 levels

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