A Characterization Theorem for Bounded Pair Algebras: The Unique O(1) Enclosure-Preserving Algebra over ℝ × ℝ≥0

We prove that a unique binary pair algebra exists over the carrier ℝ × ℝ≥0 satisfying six axioms (A1–A6): symmetric-interval semantics, exact enclosure preservation, exact tightness, commutativity and associativity, non-negativity of bounds, and O(1) summary-based propagation. The algebra — here called the EB pair algebra — has addition (e₁, b₁) ⊕ (e₂, b₂) = (e₁ + e₂, b₁ + b₂) and multiplication (e₁, b₁) ⊗ (e₂, b₂) = (e₁e₂, |e₁|b₂ + |e₂|b₁ + λ b₁b₂). Under A1–A6 with A3 read as exact tightness, λ = 1 is uniquely forced; this is the exact uniqueness result. The value λ = 0 arises not as a dual alternative but as the unique first-order approximation to the λ = 1 algebra under a declared relaxation of A3 (the first-order truncation of the tightness condition), agreeing with the exact algebra at O(b) and differing at O(b²). The characterization proceeds in three stages: carrier identification (exactly two coordinates are essential), algebraic elimination (admissible operations narrow to a one-parameter family), and tightness specification (exact tightness forces λ = 1; the first-order approximation is recovered under relaxation). The algebra exhibits a forced one-directional coupling: bounds observe expressed magnitudes, but expressed values are blind to bounds — a non-interlocked structure absent from bilattices, rough sets, fuzzy logic, and interval arithmetic, and the theorem explains why such algebras do not arise in those axiomatizations. We validate across 70,054 numerical tests and show that expressed-component cancellation of shared latent factors (e.g., H₀ in cosmological distance ratios) is a structural consequence, not parameter-tuned. The work frames a broader classification question for asymmetric pair algebras.