If `G_b ≡ S(B_bᵛ)` through degree `M` for every `b > 0`, then `S(P_sᵛ) · P_s ≡ 1` through degree `M`, where `P_s = eulerSer (-1) s`.

import Definitions.Def_TypeDensity_Basic
import Definitions.Def_TypeDensity_Formula_LawAlgebra
import Definitions.Def_TypeDensity_Formula_StateCombinatorics
import Definitions.Def_TypeDensity_Reciprocity_Defs
import Definitions.Def_TypeDensity_Reciprocity_Quantum
import Theorems.Thm_TypeDensity_constantCoeff_logSer
import Theorems.Thm_TypeDensity_expSer_logSer
import Theorems.Thm_TypeDensity_pPow_mul_pPow
import Theorems.Thm_TypeDensity_constantCoeff_clusterSer
import Theorems.Thm_TypeDensity_scalar_one
import Theorems.Thm_TypeDensity_plethSum_moebiusSum
import Theorems.Thm_TypeDensity_coeff_recipSer
import Theorems.Thm_TypeDensity_recipSer_mul
import Theorems.Thm_TypeDensity_recipSer_recipSer
import Theorems.Thm_TypeDensity_dilate_mul
import Theorems.Thm_TypeDensity_dilate_codilate
import Theorems.Thm_TypeDensity_recipSer_dilate
import Theorems.Thm_TypeDensity_SCongr_dilate
import Theorems.Thm_TypeDensity_SCongr_symm
import Theorems.Thm_TypeDensity_SCongr_trans
import Theorems.Thm_TypeDensity_SCongr_mul
import Theorems.Thm_TypeDensity_SCongr_of_mul_right
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
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.PowerSeries.Expand
import Mathlib.RingTheory.PowerSeries.Log
set_option autoImplicit false

open TypeDensity

theorem TypeDensity.dilate_recipSer_eulerSer_mul_scongr_one (M s : ℕ) (hs : 0 < s) (hS : ∀ a : ℕ, 0 < a →
    SCongr M (eulerSer 0 a) (dilate (recipSer (clusterSer a)))) :
    SCongr M (dilate (recipSer (eulerSer (-1) s)) * eulerSer (-1) s) 1 := by sorry