 149/-! ## 4. Trap characterization -/
 150
 151/-- Every prime divisor of `m` lies in residue class `1 mod 3`. -/
 152def AllPrimeFactorsOneModThree (m : ℕ) : Prop :=
 153  ∀ p : ℕ, Nat.Prime p → p ∣ m → p % 3 = 1
 154
 155/-- The residual trapped class after the explicit formulas:
 156`n ≡ 1 mod 24`, and both ledger sources are built only from primes
 157`1 mod 3`. -/
 158def ResidualTrap (n : ℕ) : Prop :=
 159  1 < n ∧
 160  n % 24 = 1 ∧
 161  AllPrimeFactorsOneModThree n ∧
 162  AllPrimeFactorsOneModThree ((n + 3) / 4)
 163
 164/-! ## 5-10. Missing engines as explicit first-principles targets -/
 165
 166/-- A gate has enough phase support if its finite quotient sees a
 167non-identity phase in the divisor ledger.  This is intentionally abstract:
 168the next proof must instantiate it from prime phase distribution. -/
 169def GateHasPhaseSupport (n c : ℕ) : Prop :=
 170  ∃ _ : ResidualPhaseQuotient c, n = n
 171
 172/-- Prime phase equidistribution: trapped ledgers are eventually separated
 173by an admissible finite quotient. -/
 174structure PrimePhaseEquidistributionEngine : Prop where
 175  phase_support :
 176    ∀ n : ℕ, ResidualTrap n →
 177      ∃ c : ℕ, AdmissibleHardGate c ∧ GateHasPhaseSupport n c
 178
 179/-- Effective bounded search: the separating gate can be chosen below a
 180specific bound. -/
 181structure BoundedSearchEngine : Type where
 182  bound : ℕ → ℕ

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