| import OAI.Combinatorics.BinaryEdit.SparseActivity |
| import OAI.Combinatorics.BinaryEdit.FiniteActivityBridge |
|
|
| namespace OAI |
|
|
| /-! |
| # Finite-tree spectral obstruction |
|
|
| The sparse/dense counting and Fourier-mass estimates give the spectral |
| bound used in `prop:finite-lower` for the cyclic leaf coordinates and |
| cheap translations. |
| -/ |
|
|
| noncomputable section |
|
|
| namespace TreeEdit.BinaryLower.FiniteActivity |
|
|
| open Classical |
| open scoped BigOperators |
|
|
| / |
| manuscript instantiate `h=b=k`, `m=20*k`, and `C=(2*J/T)*M`. The conclusion |
| then reproduces `eq:finite-root-upper` before the final numerical estimate. -/ |
| theorem finite_tree_spectral_bound {A σ : Type*} |
| [AddCommGroup A] [Fintype A] [Fintype σ] [Nonempty σ] |
| (m b h : ℕ) (hm : m ≠ 0) (hb : 0 < b) |
| (w : List (Fin m) → A) |
| (Q : (Fin h → Fin m) → ℕ) [∀ f, NeZero (Q f)] |
| (φ : ∀ f, ZMod (Q f) →+ A) |
| (hgen : ∀ f : Fin h → Fin m, w (List.ofFn f) = φ f 1) |
| (H : AddChar A ℂ → ℝ) (hH : ∀ χ, 0 ≤ H χ) |
| (shift : InternalNode (Fin m) h → σ → A) (M C : ℝ) |
| (hleaf : ∀ f (z : ZMod (Q f)), |
| (∑ χ, ‖1 - χ (φ f z)‖ ^ 2 * H χ) ≤ M / (m : ℝ) ^ h) |
| (hdetect : ∀ v χ, characterSparse w b v χ → |
| (Fintype.card σ : ℝ) ≤ ∑ s, ‖1 - χ (shift v s)‖ ^ 2) |
| (hcost : ∀ v s, |
| (∑ χ, ‖1 - χ (shift v s)‖ ^ 2 * H χ) ≤ C / (m : ℝ) ^ (v.1 : ℕ)) : |
| (∑ χ : AddChar A ℂ, ‖1 - χ (w [])‖ ^ 2 * H χ) ≤ |
| 4 * ((h : ℝ) * C + M / (2 * (b : ℝ) ^ h)) := by |
| classical |
| apply TreeEdit.BinaryLower.TreeDistortion.FiniteMass.finite_spectral_root_bound Q φ H (w []) |
| (characterSparse w b) shift |
| (fun v => C / (m : ℝ) ^ (v.1 : ℕ)) |
| (fun _f => M / (m : ℝ) ^ h) M ((h : ℝ) * C) ((b : ℝ) ^ h) |
| · exact pow_pos (Nat.cast_pos.mpr hb) _ |
| · exact hH |
| · intro χ hr hn |
| exact character_dense_of_leaf_generators w b h (fun f => φ f 1) χ hgen hr hn |
| · exact hleaf |
| · exact le_of_eq (leaf_budget m h hm M) |
| · exact hdetect |
| · exact hcost |
| · exact le_of_eq (internal_node_budget m h hm C) |
|
|
| end TreeEdit.BinaryLower.FiniteActivity |
|
|
| end |
|
|
| end OAI |
|
|
