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A new theoretical approach is described to quantify position-domain integrity risk for cycle ambiguity resolution problems in satellite-based navigation systems. It is typically conservatively assumed that all incorrectly fixed cycle ambiguities cause hazardously large position errors. While simple and practical, this conservative assumption can unnecessarily limit navigation availability for applications with stringent requirements for accuracy and integrity. In response, a new method for calculating the integrity risk for carrier phase navigation algorithms is developed. In this method we evaluate the impact of incorrect fixes in the position domain and define tight upper bounds on the resulting navigation integrity risk. Furthermore, a mechanism to implement this method with partially-fixed cycle ambiguity vectors is also derived. The improvement in navigation availability using the new method is quantified through covariance analysis performed over a range of error model parameters.