Both actual introspection measurements reach terminal form

Alice's false-role block is followed by Bob's true-role block. The second block leaves the completed Alice measurement unchanged. All its primitive inputs are transported from the original strategy, and a single numerical recurrence accounts for the full 2 * ℓ replacements.

Stated in a bipartite model (Phase 4 of planning/mipco-track.md): Alice's block returns a model N₁ over the second player's algebra, Bob's block, run on N₁.Ξ, a model N₂ over N₁'s first player's algebra, and the final model is N₂.Ξ.swap, which reduces to the original one. The matrix version's product state introTwoSidedState has no counterpart.

theorem MIPRE.Introspection.adaptiveFailureBudget_add (r : ℕ) (edges z h initial : ℝ) (m n : ℕ) : adaptiveFailureBudget r edges z h initial (m + n) = adaptiveFailureBudget r edges z h (adaptiveFailureBudget r edges z h initial m) n

Restarting the failure recurrence is exactly continuing it.

theorem MIPRE.Introspection.TypedEstimates.exists_intro_two_sided_iteration {PauliType : Type u_1} {PauliAnswer : Type u_2} {κ : Type u_3} {F ι A : Type} [Fintype PauliType] [DecidableEq PauliType] [Fintype PauliAnswer] [Field F] [Fintype F] [DecidableEq F] [Algebra (ZMod 2) F] [Fintype ι] [DecidableEq ι] [Fintype κ] [DecidableEq κ] [Fintype A] {ℓ : ℕ} {𝒞 𝒜 ℬ : Type u} [Ring 𝒞] [StarRing 𝒞] [Algebra ℂ 𝒞] [Ring 𝒜] [StarRing 𝒜] [Algebra ℂ 𝒜] [StarModule ℂ 𝒜] [PartialOrder 𝒜] [StarOrderedRing 𝒜] [StarProper 𝒜] [Ring ℬ] [StarRing ℬ] [Algebra ℂ ℬ] [StarModule ℂ ℬ] [PartialOrder ℬ] [StarOrderedRing ℬ] [StarProper ℬ] (E : PauliType → PauliType → Bool) (X Z : PauliType) (P : PauliType → CL.CLFun (ZMod 2) κ 3) (L : Bool → CL.CLFun F ι ℓ) (projectPauli : PauliAnswer → ι → F) (D : (ι → F) → (ι → F) → A → A → Bool) (DP : PauliType → PauliType → (κ → ZMod 2) → (κ → ZMod 2) → PauliAnswer → PauliAnswer → Bool) (Ξ : BipartiteModel 𝒞 𝒜 ℬ) (hΞ : ‖Ξ.ψ‖ = 1) (MA : CL.Detyping.Question (QuestionType PauliType ℓ) κ → POVMIn (ParsedAnswer (ι → F) A PauliAnswer) (Matrix (ι → F) (ι → F) 𝒜)) (MB : CL.Detyping.Question (QuestionType PauliType ℓ) κ → POVMIn (ParsedAnswer (ι → F) A PauliAnswer) (Matrix (ι → F) (ι → F) ℬ)) (hMA : ∀ (q : CL.Detyping.Question (QuestionType PauliType ℓ) κ), IsPVMIn (MA q).op) (hMB : ∀ (q : CL.Detyping.Question (QuestionType PauliType ℓ) κ), IsPVMIn (MB q).op) (q : κ → ZMod 2) (hq : ∀ (z : κ → ZMod 2), (P Z).eval z = q) (hL : ∀ (w : Bool), (L w).SupportedOn Finset.univ) {initial η δ : ℝ} (hi0 : 0 ≤ initial) (hη0 : 0 ≤ η) (hδ0 : 0 ≤ δ) (hfail : 1 - (Ξ.reg (ι → F)).povmValue (parsedGame E X Z P L projectPauli D DP) MA MB ≤ initial) (hZA : introAliceZError projectPauli Z q Ξ MA ≤ η) (hZB : introBobZError projectPauli Z q Ξ MB ≤ η) (hhideA : ∀ (j : Fin ℓ), hidingAliceError L false ⋯ Ξ MA j ≤ δ) (hhideB : ∀ (j : Fin ℓ), hidingBobError L true ⋯ Ξ MB j ≤ δ) (hi : initial ≤ adaptiveSmallThreshold ℓ (↑(TypeGraph.edges E X Z ℓ).card) (2 * ℓ)) (hη : η ≤ adaptiveSmallThreshold ℓ (↑(TypeGraph.edges E X Z ℓ).card) (2 * ℓ)) (hδ : δ ≤ adaptiveSmallThreshold ℓ (↑(TypeGraph.edges E X Z ℓ).card) (2 * ℓ)) : ∃ (N₁ : ModelOver ℬ) (N₂ : ModelOver N₁.𝒜) (MAN : CL.Detyping.Question (QuestionType PauliType ℓ) κ → POVMIn (ParsedAnswer (ι → F) A PauliAnswer) (Matrix (ι → F) (ι → F) N₁.𝒜)) (MBN : CL.Detyping.Question (QuestionType PauliType ℓ) κ → POVMIn (ParsedAnswer (ι → F) A PauliAnswer) (Matrix (ι → F) (ι → F) N₂.𝒜)) (x : IntroPrefixInvariant (L false) ℓ (POVMIn.map introspectPair (MAN (QuestionType.introspect false, 0)))) (x : IntroPrefixInvariant (L true) ℓ (POVMIn.map introspectPair (MBN (QuestionType.introspect true, 0)))), ‖N₂.Ξ.ψ‖ = 1 ∧ (∀ (t : CL.Detyping.Question (QuestionType PauliType ℓ) κ), IsPVMIn (MAN t).op) ∧ (∀ (t : CL.Detyping.Question (QuestionType PauliType ℓ) κ), IsPVMIn (MBN t).op) ∧ 1 - (N₂.Ξ.swap.reg (ι → F)).povmValue (parsedGame E X Z P L projectPauli D DP) MAN MBN ≤ adaptiveFailureBudget ℓ (↑(TypeGraph.edges E X Z ℓ).card) η δ initial (2 * ℓ) ∧ N₂.Ξ.swap.POVMReduces Ξ

Two concrete projective measurement families with both terminal prefix invariants, from actual game success and fixed primitive errors, in a model reducing to the original one. No future measurement, structural invariant, or stage-success premise is assumed.