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.

The unshifted lower Laplacian bound excludes a boundary degree

Statement

Fix $γ > 1$ . For each positive integer $n$ , put $μ = γ \log n$ and $p = μ / n$ , and consider only $n$ for which $p \leq 1$ . Let $c^{-} \in (- 1, 0)$ be the unshifted root of $γ ((1 + c) \log (1 + c) - c) = 1$ . Define the scalar function $I (x) = x \log x - x + 1$ for $x > 0$ ; thus $I (1 + c) = h (c)$ for $c > - 1$ . Set $a = 1 + c^{-} \in (0, 1)$ and $k_{n} = ⌈ a μ - p ⌉ - 1$ . In the loop-free graph $G (n, p)$ , any vertex of degree $k_{n}$ prevents $c^{-} μ I ⪯ Δ_{L}$ , where $Δ_{L} = D - A - μ I + p J$ . Consequently it prevents the full $(n, μ, α, c^{-}, c^{+})$ -expander predicate, for every $α, c^{+}$ . Moreover $k_{n} = a μ + O (1)$ as $n \to \infty$ at fixed $γ$ , and $γ I (a) = 1$ .

Facts & Assumptions

[F1]

The expander predicate contains the lower quadratic-form inequality $c^{-} μ I ⪯ Δ_{L}$ , with $Δ_{L} = D - A - μ I + p J$ (Two-sided spectral expander — Definition 1.9).

[F2]

The Erdős–Rényi graph is loop-free, so $A_{v v} = 0$ and $D_{v v} = d (v)$ (Erdős–Rényi random graph and high probability).

[F3]

The function $h (c) = (1 + c) \log (1 + c) - c$ on $c > - 1$ and the unshifted root satisfy $c^{-} \in (- 1, 0)$ and $γ h (c^{-}) = 1$ (Degree large-deviation rate function and roots).

[F4]

At fixed $γ$ , $p = γ \log n / n \to 0$ (The logarithm grows more slowly than every positive real power).

Proof

Given: The parameters, root and integer $k_{n}$ in the Statement.

Goal: Check the exact original diagonal obstruction including the centering term.

1.1F1F2givenalgebra

Evaluate the lower quadratic-form inequality in [F1] on the coordinate vector $e_{v}$ of norm one. By [F2] it requires $d (v) - μ + p \geq c^{-} μ$ , equivalently $d (v) \geq a μ - p$ . The integer $k_{n} = ⌈ a μ - p ⌉ - 1$ is strictly smaller than $a μ - p$ , including when the latter is itself an integer. Thus $d (v) = k_{n}$ contradicts this necessary inequality.

2.1F1F3F4givenstep 1.1algebra∎

The defining property of the ceiling gives $- 1 \leq k_{n} - (a μ - p) < 0$ , so $∣ k_{n} - a μ ∣ \leq 1 + p$ . By [F4], $p \to 0$ at fixed $γ$ ; hence every sufficiently large positive integer $n$ is admissible and $k_{n} = a μ + O (1)$ . The inline definition gives $I (a) = I (1 + c^{-}) = h (c^{-})$ , so [F3] yields $γ I (a) = 1$ . These facts prove every assertion, and the unused adjacency bound cannot remove this necessary obstruction.

Depends on

Used by · 1

The exact unshifted expansion event cannot have polynomial failure decay

Dependency tree · 2 levels

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