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). v2.0.0 (2026-04-23) — Editor-response revision. Astronomy-first abstract rewrite per editor guidance; DOI mesh retargeted to the priority-lock chain (concept DOI 10.5281/zenodo.19676236); body cites updated to the peer companion set (SIT 10.5281/zenodo.19692434, Partition 10.5281/zenodo.19655951, Decomposition v2 10.5281/zenodo.19687314); cover letter r2 bundled; marked-changes LaTeX + PDF included for reviewer diff. YouTube: @EricDMartin @CarrierSet @HubbleTension Updated 2026-05-28