A single-branch state `S(a;

Proved

TypeDensity.self_patterns_of_single_state

by asving · Oct 16, 2026 · Mathlib 0df444a (Lean v4.33.1)

type-density

A single-branch state S(a; c) with c ≥ 2 has exactly one self pattern (of multiplicity d = c, with the state itself as only child), whose tangent count at q is q.

Preamble

import Definitions.Def_TypeDensity_Basic
import Definitions.Def_TypeDensity_Formula_LawAlgebra
import Definitions.Def_TypeDensity_Formula_StateCombinatorics
import Theorems.Thm_TypeDensity_expression_prod_eval
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
open scoped Classical

Formal statement

theorem TypeDensity.self_patterns_of_single_state (a c q : ℕ) (ha : 0 < a) (hc : 2 ≤ c) :
    (∀ P : TangentPattern (State.single a c).degree, Allowed (State.single a c) P →
        SelfPattern (State.single a c) P →
          P.d.val = c ∧ children (State.single a c) P = [(1, State.single a c)]) ∧
      (((allowedPatterns (State.single a c)).toList.filter (SelfPattern (State.single a c))).map
        fun P ↦ (tangentCount (State.single a c) P).eval q).sum = q := by sorry

Source

TypeDensity/Formula/StateCombinatorics.lean#L177-L191

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