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f0f05de | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 | 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
/-- The complete sparse/dense spectral estimate on a finite tree. For the
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
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