**A concrete universe exists**: choosing child data for every cell of every concrete member yields a surface universe indexed by concrete members that satisfies the termination hypotheses.

import Definitions.Def_TypeDensity_Basic
import Definitions.Def_TypeDensity_Blowup_Directions
import Definitions.Def_TypeDensity_Measure_AdicCylinders
import Definitions.Def_TypeDensity_Formula_LawAlgebra
import Definitions.Def_TypeDensity_Measure_HenselProducts
import Definitions.Def_TypeDensity_Surfaces_Lengths
import Definitions.Def_TypeDensity_Surfaces_PowerSeries
import Definitions.Def_TypeDensity_Formula_FiniteFieldCounts
import Definitions.Def_TypeDensity_Formula_StateCombinatorics
import Definitions.Def_TypeDensity_Formula_PatternCounts
import Definitions.Def_TypeDensity_Unramified_Extension
import Definitions.Def_TypeDensity_Blowup_Charts
import Definitions.Def_TypeDensity_Algebra_AlgebraType
import Definitions.Def_TypeDensity_Measure_PrimitiveLocus
import Definitions.Def_TypeDensity_Algebra_FactorInvariant
import Definitions.Def_TypeDensity_Surfaces_TruncatedSeries
import Definitions.Def_TypeDensity_Blowup_TwoChartCurveData
import Definitions.Def_TypeDensity_Surfaces_ConcreteAtlas
import Definitions.Def_TypeDensity_Recursion_Members
import Definitions.Def_TypeDensity_Measure_MemberCharts
import Definitions.Def_TypeDensity_Recursion_Termination
import Definitions.Def_TypeDensity_Recursion_OrderDefect
import Definitions.Def_TypeDensity_Assembly_NullBridges
import Definitions.Def_TypeDensity_Assembly_TypeEvents
import Definitions.Def_TypeDensity_Assembly_MonicCells
import Definitions.Def_TypeDensity_Blowup_DivisorPoints
import Definitions.Def_TypeDensity_Measure_MemberBoxes
import Definitions.Def_TypeDensity_Recursion_ChildMembers
import Definitions.Def_TypeDensity_Recursion_RootCellLaw
import Theorems.Thm_TypeDensity_closure_finite_ascent
import Theorems.Thm_TypeDensity_algebra_type_unique
import Theorems.Thm_TypeDensity_chartBase_x_eq
import Theorems.Thm_TypeDensity_fixed_tangent_cell_generator_coefficients
import Theorems.Thm_TypeDensity_self_patterns_of_single_state
import Theorems.Thm_TypeDensity_strict_transform_ideal_of_generator
import Theorems.Thm_TypeDensity_algebra_type_children_exists
import Theorems.Thm_TypeDensity_pi_quotient_span_linearEquiv
import Theorems.Thm_TypeDensity_member_cell_box_measurable
import Theorems.Thm_TypeDensity_member_cell_box_measure
import Theorems.Thm_TypeDensity_member_children_degree_sum
import Theorems.Thm_TypeDensity_cell_child_data_exists
import Theorems.Thm_TypeDensity_concrete_member_cells_disjoint
import Theorems.Thm_TypeDensity_concrete_member_terminal_type
import Theorems.Thm_TypeDensity_unramified_generic_fibre_ascent
import Theorems.Thm_TypeDensity_concrete_member_cell_measurable
import Theorems.Thm_TypeDensity_concrete_member_stateSet_measurable
import Theorems.Thm_TypeDensity_concrete_member_curve_order_free
import Theorems.Thm_TypeDensity_concrete_type_add_of_index_map
import Theorems.Thm_TypeDensity_etale_invariant_children
import Theorems.Thm_TypeDensity_concrete_member_divisor_count
import Theorems.Thm_TypeDensity_cell_child_bijOn
import Theorems.Thm_TypeDensity_cell_member_index_map
import Theorems.Thm_TypeDensity_concrete_member_cells_cover
import Theorems.Thm_TypeDensity_concrete_member_volume
import Theorems.Thm_TypeDensity_algebra_type_unramified_descent
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.Topology.Algebra.MvPolynomial
import Mathlib.FieldTheory.Finite.GaloisField
import Mathlib.RingTheory.AdicCompletion.Algebra
import Mathlib.RingTheory.RegularLocalRing.Defs
import Mathlib.Algebra.Group.Pi.Units
import Mathlib.Order.CompletePartialOrder
import Mathlib.RingTheory.Henselian
import Mathlib.RingTheory.Polynomial.Resultant.Basic
import Mathlib.RingTheory.SimpleRing.Principal
import Mathlib.FieldTheory.Finite.Extension
import Mathlib.RingTheory.AdicCompletion.Noetherian
import Mathlib.RingTheory.MvPowerSeries.Equiv
import Mathlib.RingTheory.PicardGroup
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.AdicCompletion.Completeness
import Mathlib.Algebra.GroupWithZero.Submonoid.CancelMulZero
import Mathlib.Algebra.Module.StablyFree.Basic
import Mathlib.NumberTheory.Padics.RingHoms
import Mathlib.Algebra.Azumaya.Basic
import Mathlib.RingTheory.AdicCompletion.RingHom
import Mathlib.RingTheory.TotallySplit
import Mathlib.RingTheory.AdicCompletion.LocalRing
import Mathlib.RingTheory.KrullDimension.NonZeroDivisors
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Order.Ring.Star
import Mathlib.Data.Int.Star
import Mathlib.LinearAlgebra.Dimension.Torsion.Finite
set_option autoImplicit false

open TypeDensity
open scoped BigOperators ENNReal

theorem TypeDensity.concrete_universe_exists : ConcreteUniverse := by sorry