Shared-factor overcounting in ratio observables: an algebraic characterization

When a ratio observable inherits uncertainty from a shared scale factor, symmetric summary-state propagation re-encodes the single source as two independent contributions, producing an overcounting of order √2·σ/μ. This paper prices that cost, proves it is structurally unavoidable under symmetric coupling, and identifies the unique O(1) alternative. A characterization theorem shows that exactly one bounded pair algebra over ℝ × ℝ≥0 satisfies enclosure, tightness, associativity, and constant-time complexity. That algebra reallocates the cost: per-operation bounds widen, but the shared factor cancels on the expressed rail, eliminating the system-level overcounting without covariance bookkeeping. The Hubble tension provides a concrete invoice: a pure H0 frame shift produces 4.89σ in H0 space but exactly 0σ in the ratio coordinate ξ = dc/dH, while a physical matter-density difference survives at ≈1.0σ, confirming that the reallocation preserves physical content. Manuscript: MN-26-1108-L (MNRAS Letters, submitted).