What question did this study set out to answer?

The aim is to characterize and address the overcounting of uncertainty in ratio observables due to shared factors.

April 25, 2026Open Access

Shared-factor overcounting in ratio observables: an algebraic characterization

Key Points

The aim is to characterize and address the overcounting of uncertainty in ratio observables due to shared factors.
Developed a characterization theorem for bounded pair algebra over ℝ × ℝ≥0.
Proved the structural inevitability of overcounting under symmetric coupling.
Identified an alternative algebra that reallocates costs without covariance bookkeeping.
The proposed algebra eliminates system-level overcounting while preserving physical content.
A frame shift in H0 produced 4.89σ in H0 space but 0σ in the ratio coordinate, indicating successful cancellation.
Demonstrated that physical matter-density differences survive at approximately 1.0σ.

Abstract

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.

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