The explicit endpoint transmission block

The endpoint pencil determines a decreasing block in the transmission permutation. This is the algebraic input to the endpoint inversion bound.

theorem Bananas.exists_endpoint_transmission_block {g k : ℕ} (B : Banana g) (hsub : AllSubmodular (mark (Utilities.Certificate.SubdivisionGraph.Spec.graph B) (leftEndpoint B) (rightEndpoint B))) (hk : TorsionWitness (mark (Utilities.Certificate.SubdivisionGraph.Spec.graph B) (leftEndpoint B) (rightEndpoint B)) k) : ∃ (τ : ℤ → ℤ), IsTransmissionPermutation (mark (Utilities.Certificate.SubdivisionGraph.Spec.graph B) (leftEndpoint B) (rightEndpoint B)) (g • oneChip (rightEndpoint B)) τ ∧ IsKAffine k τ ∧ ∀ b ≤ g, τ ↑b = ↑(g - b)

theorem Bananas.endpoint_affine_period_gt_genus {g k : ℕ} {τ : ℤ → ℤ} (hk : 0 < k) (hAffine : IsKAffine k τ) (hBlock : ∀ b ≤ g, τ ↑b = ↑(g - b)) : g < k

The decreasing endpoint block forces every affine period to be larger than the genus. In particular its choose (g+1) 2 ordinary inversions lie in distinct period classes.