For `q, c ≥ 2`, the return weight `q · q^(-tri c)` lies in `(0, q⁻²]`, in particular below `1`.

import Definitions.Def_TypeDensity_Basic
import Definitions.Def_TypeDensity_Formula_LawAlgebra
import Definitions.Def_TypeDensity_Formula_StateCombinatorics
import Mathlib.FieldTheory.RatFunc.Basic
import Mathlib.MeasureTheory.Measure.Haar.Unique
import Mathlib.NumberTheory.ArithmeticFunction.Moebius
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.IntegralClosure.IntegralRestrict
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.RingTheory.RamificationInertia.Basic
import Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
import Mathlib.FieldTheory.Finite.GaloisField
import Mathlib.RingTheory.AdicCompletion.Algebra
import Mathlib.RingTheory.RegularLocalRing.Defs
set_option autoImplicit false

open TypeDensity
open scoped BigOperators ENNReal

theorem TypeDensity.return_weight_bounds (q c : ℕ) (hq : 2 ≤ q) (hc : 2 ≤ c) :
    0 < (returnWeight c).eval q ∧ (returnWeight c).eval q ≤ (q : ℚ)⁻¹^2 ∧
      (returnWeight c).eval q < 1 := by sorry