Explain detector-pattern dependence of error responses

Derive a comprehensive explanation for why error-response coefficients, particularly those associated with Pauli errors and correlated leakage, vary across successful detector patterns in seed-state generators with multiple heralding patterns.

Background

Several SSGs with multiple successful detector patterns exhibit substantially different output-error responses depending on the particular pattern that is measured. The paper gives the example of source distinguishability producing widely varying Pauli-Z response coefficients across detector patterns, and notes that correlated leakage responses can also vary significantly.

Although the framework calculates these pattern-specific coefficients and could support decisions to discard especially error-prone nominally successful patterns, the underlying reason for the observed variations is not established. A general explanation would help optimize heralding rules and quantify the trade-off between success probability and output-state quality.