1. Let's restate the problem: We want to predict the value of a smaller time frame within a larger time frame based on the smaller frame's value before the larger frame ends, considering overlapping time frames and exceptions. 2. This is a problem of time series analysis and conditional prediction, often addressed using concepts like conditional expectation and correlation in statistics. 3. A common mathematical tool is the concept of conditional expectation: $$E[X_s \mid X_l]$$ where $X_s$ is the value of the smaller time frame and $X_l$ is the value of the larger time frame. 4. If the smaller time frame's value is small before the larger time frame ends, and if the two are positively correlated, then the expected value of the smaller time frame can be predicted to be positive or negative accordingly. 5. However, overlapping time frames introduce dependencies that complicate direct prediction. One approach is to model the joint distribution of $X_s$ and $X_l$ and use Bayesian inference or regression models to predict $X_s$ given partial information about $X_l$. 6. There is no single "rule" but rather statistical models such as autoregressive models, Kalman filters, or machine learning models that can incorporate overlapping time frames and exceptions to improve prediction accuracy. 7. In summary, the mathematical framework involves conditional expectation and correlation, but practical prediction requires statistical or machine learning models tailored to the data and overlap structure. Final answer: There is no simple closed-form rule, but conditional expectation and statistical modeling provide the mathematical basis for predicting smaller time frame values within larger overlapping time frames.