Lax Lean Archive

The Explicit Subcubic Family Expands

Lax920634.ExplicitExpansion · concepts/Lax920634/ExplicitExpansion.lean · lax-920634

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    Natural Language Statement

    Lemma

    Put D=1205760000D=120^{5760000} and d=D2d=D^2. Every specified graph HkH_k has edge expansion at least 1/(8d)1/(8d), uniformly in the level.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

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    1import Lax920634.ZigZagConstruction
    2
    3/-!
    4---
    5title: The Explicit Subcubic Family Expands
    6type: lemma
    7---
    8Put D=1205760000D=120^{5760000} and d=D2d=D^2. Every specified graph HkH_k has edge
    9expansion at least 1/(8d)1/(8d), uniformly in the level.
    10
    11# Formalization notes
    12
    13The constant depends only on the fixed seed degree, not on the graph order.
    14The proof combines the zig-zag spectral bound with path replacement and
    15checks the canonical vertex numbering.
    16-/
    17
    18namespace Lax920634.ExplicitExpansion
    19
    20open ZigZagConstruction EdgeExpansion
    21
    22/-- The specified deterministic family has a uniform positive edge-expansion constant. -/
    23axiom expansion (k : ℕ) : HasEdgeExpansionAtLeast
    24 (graph k) (1 / (8 * (degree : ℚ)))
    25
    26end Lax920634.ExplicitExpansion
    27
    Show Proof
    Formalization notes

    The constant depends only on the fixed seed degree, not on the graph order. The proof combines the zig-zag spectral bound with path replacement and checks the canonical vertex numbering.

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