Structural limits of single-barrier reform in algorithmic recourse: a formal series-system model with implications for digital health

Algorithmic decision systems mediate access to healthcare, credit, employment and housing, yet individuals who experience adverse decisions face multi-stage barriers when seeking recourse. We formalize these barriers as a series-structured system with 11 empirically parameterized stages across three layers (data integration, data accuracy and institutional access) and prove that single-barrier interventions are bounded by baseline system success. Under baseline parameterization derived from federal datasets and peer-reviewed algorithmic audit studies, end-to-end recourse probability is 0.0018%. Removing any single barrier yields negligible improvement (<0.02%). Factorial decomposition reveals that the three-way cross-layer interaction accounts for 87.6% of achievable improvement, illustrated by Shapley attribution, Sobol sensitivity analysis and bootstrap resampling (n = 1,000) within prespecified perturbation ranges. Introducing dependence among the stage pass-events with a Gaussian-copula analysis (latent correlation up to {rho} = 0.5), while preserving each marginal pass probability, leaves the comparative conclusion--single-layer reform yields far less than coordinated reform--intact, while a repeated-attempt topology attenuates it, delimiting the models domain of validity. These findings are properties of the modeled system rather than empirical measurements of real recourse pathways; they yield testable hypotheses for digital-health governance--chiefly that single-layer fairness interventions are structurally unlikely to produce large absolute gains--pending validation against observed recourse data. Author summaryWhen an algorithm makes an adverse decision about you--denying a loan, filtering your job application, or miscategorizing your health risk--what does it take to correct it? We built a formal mathematical model of the 11 barriers a person must clear to obtain recourse, from detecting the error to accessing legal resources, and studied its structure. Calibrated with data from federal agencies and one healthcare audit--a provisional, cross-domain calibration--the model puts the probability of clearing all barriers below 2 in 100,000, and in the model no single-barrier fix raises it by more than 0.02%. This follows from the models multiplicative structure: barriers compound rather than add. Within the model, the three-way interaction across data integration, data accuracy and institutional access accounts for 87.6% of the achievable improvement. These are properties of the model rather than measurements of real recourse; they generate a testable hypothesis--that coordinated, multi-layer reform may outperform isolated single-layer fixes--which remains to be validated against observed recourse data.