Choosing an astigmatic correction by expected loss: different goals can favour different options

Purpose: To examine whether choosing a correction that minimizes expected squared residual also reduces the probability of exceeding a residual-cylinder target. Methods: We first used an exact two-option example to distinguish mean cylinder, mean squared cylinder and threshold exceedance. We then evaluated 10,000 preserved synthetic scenarios using fixed local menus of 25 corneal-plane correction identifiers. The primary menu used neighbouring 10° axes; a 5° menu was a sensitivity analysis. A common additive-error control preserved the expected-squared-loss ranking. With additional normally distributed physical rotation, the candidate correction was selected analytically to minimize expected squared residual. Independent evaluation states estimated mean cylinder, threshold exceedance and expected excess. All probabilities were conditional on assumed error laws. Results: Under the primary law, with additive component SD 0.25 D and rotation SD 3°, expected-squared-loss selection changed 289/10,000 choices in the 10° menu. Mean expected squared residual fell from 0.322661 to 0.322421 D². However, mean probability of residual above 0.50 D rose from 40.3291% to 40.3568%, a difference of 0.02772 percentage points (paired Monte Carlo SE 0.00075). Probability above 1.00 D fell from 6.0759% to 6.0544%, and expected excess above 0.50 D fell from 0.107171 to 0.107054 D. The zero-rotation control changed no choices. Larger assumed rotation increased the squared-loss gain but also increased the 0.50-D exceedance penalty. Conclusions: A correction can reduce expected squared error while slightly increasing failure to meet a particular target. Calculation tools should state the objective and show frequency and magnitude together. These synthetic results do not establish calibrated patient probabilities or a clinical benefit. This preprint and its companion study use the same bank of 10,000 synthetic scenarios; they are not independent clinical replications. The assumptions are not calibrated patient probabilities and no clinical benefit is established. The deposit provides the full manuscript and supplement in Markdown, editable DOCX and PDF, and a transcription of the main aggregate tables. The complete input fixture, analysis code, execution records and scenario-level outputs are not supplied; the full numerical census cannot be independently rerun from this deposit alone.