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The retention tool models four populations with entered transition assumptions. At default day-one retention point four, the source active-persistence factor is point four times point eight five, or point three four. The initial ten thousand new users move to three thousand four hundred active and six thousand six hundred inactive. It then adds five thousand daily new installs. Its daily-active metric includes both newly added and active users, producing eight thousand four hundred out of fifteen thousand cumulative installs on day one. These bars share point zero two pixels per person: sixty eight for active and one hundred for new. They describe modeled counts, not measured app activity or a real conversion rate. On day two, the five thousand new plus three thousand four hundred active from the previous day contribute eight thousand four hundred times point three four, or two thousand eight hundred fifty six active users. The six thousand six hundred inactive users contribute another one hundred thirty two through the point zero two resurrection assumption. Active therefore totals two thousand nine hundred eighty eight. Adding the next five thousand new produces seven thousand nine hundred eighty eight daily-active users. The two bars use point zero five pixels per person, measuring one hundred forty two point eight and six point six for the two active contributions. Churn receives five percent of the inactive count, or three hundred thirty. These are deterministic expectation values rather than random individual behavior. The panel called steady equilibrium is calculated separately from the four-state transition loop. It sums a power-law retention curve across three hundred sixty five days, multiplies by daily installs, then multiplies by one plus twice the resurrection rate. The curve exponent uses day-one and day-thirty inputs. The day-seven input does not enter either shown equation. The two bars illustrate the default trajectory horizon one hundred eighty and the separate three hundred sixty five-day sum at point eight pixels per day, measuring one hundred forty four and two hundred ninety two. This is not a solved infinite-horizon Markov equilibrium. Offline Math and D three dependencies must be checked before claiming working charts or controls. The model is an unvalidated planning exercise, not observed user retention.